This is a preprint of: “A quasiperiodically forced skew-product on the cyclinder without fixed- curves”, Llu´ıs Alsed` a, Francesc Ma˜ nosas, Leopoldo Morales, Nonlinear Anal., vol. 145, 199–263, 2016. A QUASIPERIODICALLY FORCED SKEW-PRODUCT ON THE CYLINDER WITHOUT FIXED-CURVES LLU´IS ALSEDA, ` FRANCESC MANOSAS, ˜ AND LEOPOLDO MORALES Abstract. In [1] the Sharkovski˘ı Theorem was extended to periodic orbits of strips of quasiperiodic skew products in the cylinder. In this paper we deal with the following natural question that arises in this setting: Does Sharkovski˘ı Theorem holds when restricted to curves instead of general strips? We answer this question in the negative by constructing a counterexample: We construct a map having a periodic orbit of period 2 of curves (which is, in fact, the upper and lower circles of the cylinder) and without any invariant curve. In particular this shows that there exist quasiperiodic skew products in the cylinder without invariant curves. 1. Introduction We consider the coexistence and implications between periodic objects of maps on the cylinder Ω = S1 × I, of the form:     θ Rω (θ) F: −→ , x ζ(θ, x) where S1 = R/Z, I is an interval of the real line, Rω (θ) = θ + ω (mod 1) with ω ∈ R\Q and ζ(θ, x) = ζθ (x) is continuous on both variables. The class of all maps of the above type will be denoted by S(Ω). In this setting a very basic and natural question is the following: is it true that any map in the class S(Ω) has an invariant curve? In [1], the authors created an appropriate topological framework that allowed them to obtain the following extension of the Sharkovski˘ı Theorem to the class S(Ω)1. Let X be a compact metric space. We recall that a subset G ⊂ X is residual if it contains the intersection of a countable family of open dense subsets in X. In what follows, π : Ω −→ S1 will denote the standard projection from Ω to the circle. Given a set B ⊂ S1 , for convenience we will use the following notation:  B := π −1 (B) = B × I ⊂ Ω In the particular case when B = {θ}, instead of  {θ} we will simply write  θ. Also, given A ⊂ Ω, we will denote by AB the set A ∩  B = {(θ, x) ∈ Ω : θ ∈ B and (θ, x) ∈ A}. Date: February 29, 2016. 1991 Mathematics Subject Classification. Primary: 37C55, 37C70. Key words and phrases. Quasiperiodically forced systems on the cylinder, invariant strips. The authors have been partially supported by MINECO grant numbers MTM2008-01486, MTM2011-26995-C02-01 and MTM2014-52209-C2-1-P. 1As already remarked in [1], instead of S1 we could take any compact metric space Θ that admits a minimal homeomorphism R : Θ −→ Θ such that R` is minimal for every ` > 1. However, for simplicity and clarity we will remain in the class S(Ω). 1 2 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES  In the particular case when B = {θ}, instead of Aθ we will simply write Aθ . Instead of periodic points we use objects that project over the whole S1 , called strips in [1, Definition 3.9]. A set B ⊂ Ω such that π(B) = S1 (i.e., B projects on the whole S1 ) will be called a circular set. Definition 1.1. A strip in Ω is a compact circular set B ⊂ Ω such that B θ is a closed interval (perhaps degenerate to a point) for every θ in a residual set of S1 . Given two strips A and B, we will write A < B and A ≤ B ([1, Definition 3.13]) if   there exists a residual set G ⊂ S1 , such that for every (θ, x) ∈ A G and (θ, y) ∈ B G it follows that x < y and, respectively, x ≤ y. We say that the strips A and B are ordered (respectively weakly ordered ) if either A < B or A > B (respectively A ≤ B or A ≥ B). Definition 1.2 ([1, Definition 3.15]). A strip B ⊂ Ω is called n-periodic for F ∈ S(Ω) if F n (B) = B and the image sets B, F (B), F 2 (B), . . . , F n−1 (B) are pairwise disjoint and pairwise ordered (see Figure 1 for examples). 0.9 1 0.85 0.9 0.8 0.8 0.75 0.7 0.7 0.65 0.6 0.6 0.5 0.55 0.4 0.5 0.3 0.45 0.2 0.4 0.35 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 4 111 (a) 3.28x(1 − x) + 100 cos(2πθ) (b) 3.85x(1 − x)(1 + 105 cos(2πθ)) Figure 1. In the left picture we show an example two periodic orbit of curves, and in the second we show a possible example of a three periodic orbit solid strips. To state the main theorem of [1] we need to recall the Sharkovski˘ı Ordering ([2, 3]). The Sharkovski˘ı Ordering is a linear ordering of N defined as follows: 3 Sh > 5 Sh > 7 Sh > 9 Sh > · · · Sh > 2 · 3 Sh > 2 · 5 Sh > 2 · 7 Sh > 2 · 9 Sh > · · · Sh > 4 · 3 Sh > 4 · 5 Sh > 4 · 7 Sh > 4 · 9 Sh > · · · Sh > .. . 2n · 3 Sh > 2n · 5 Sh > 2n · 7 Sh > 2n · 9 Sh > · · · Sh > .. . · · · Sh > 2n Sh > · · · Sh > 16 Sh > 8 Sh > 4 Sh > 2 Sh > 1. In the ordering Sh ≥ the least element is 1 and the largest one is 3. The supremum of the set {1, 2, 4, . . . , 2n , . . . } does not exist. Sharkovski˘ı Theorem for maps from S(Ω) ([1]). Assume that the map F ∈ S(Ω) has a p-periodic strip. Then F has a q-periodic strip for every q θ2 . The closed arc of S1 joining θ1 and θ2 in the natural direction will be denoted by [θ1 , θ2 ]. That is, ( {t (mod 1) : θ1 ≤ t ≤ θ2 } when θ1 ≤ θ2 , and [θ1 , θ2 ] = {t (mod 1) : θ1 ≤ t ≤ θ2 + 1} when θ1 > θ2 . 8 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES 1 β(θ) −1 1 φ(θ) −β(θ) −1 Figure 2. The graphs of the functions φ (in blue) and ±β in thick black. The red dashed curve is (1 − |x|)2 . The open arc of S1 joining θ1 and θ2 will be denoted by (θ1 , θ2 ) = [θ1 , θ2 ]\{θ1 , θ2 }, and is defined analogously with strict inequalities Given an arc B ⊂ S1 , Bd(B) will denote the set of endpoints of B. We will denote the open (respectively closed) ball (in S1 ) of radius δ centred at θ ∈ S1 by Bδ (θ) (respectively Bδ [θ]): Bδ (θ) = {θe ∈ S1 : dS1 (θ, θ) e < δ} = (θ − δ (mod 1), θ + δ (mod 1)), and Bδ [θ] = Bδ (θ) = {θe ∈ S : dS1 (θ, θ) 1 e ≤ δ} = [θ − δ (mod 1), θ + δ (mod 1)]. We consider the space Ω endowed the metric induced by the maximum of dS1 and the absolute value on I. That is, given (θ, x), (ν, y) ∈ Ω we set  dΩ ((θ, x), (ν, y)) := max dS1 (θ, ν), |x − y| . Then, given A ⊂ Ω we will denote the interior of A by Int(A) and diam(A) will denote the diameter of A whenever A is compact. To define the sequence of pseudo-curves that will converge to A(γ,G) we first need to construct an auxiliary family {R(`∗ )}`∈Z of compact regions in Ω and a family of compact sets {Γϕ`∗ }`∈Z such that, for every ` ∈ Z, Γϕ`∗ ⊂ R(`∗ ) and it is the restriction of a pseudo-curve generator to π(R(`∗ )). To do this we define the auxiliary functions β : [−1, 1] −→ [−1, 1] and φ : [−1, 1]\{0} −→ [−1, 1] by (see Figure 2): π β(x) := 1 − |x| and φ(x) := (1 − |x|)2 sin . x Note that −β(x) < φ(x) < β(x), for all x ∈ [−1, 1]\{0} and the graphs of −β and β intersect the closure of the graph of φ only at the points (0, −1), (0, 1), (−1, 0) and (1, 0). To define the families {R(`∗ )}`∈Z and {Γϕ`∗ }`∈Z we use the following generic boxes. A SKEW-PRODUCT WITHOUT FIXED-CURVES 9 1 a+ 2n (β ◦ ϑ−1 `∗ )(θ) a+ a a− Γϕ(`∗ ,n,α,δ,a,a+ ,a− ) `∗ − α `∗ − δ `∗ `∗ + δ `∗ + α 1 a− 2n (β ◦ ϑ−1 `∗ )(θ) Figure 3. The region R(`∗ , n, α, δ, a, a+ , a− ) is the colored filled area, delimited in the rectangle  Bδ [`∗ ] by the graphs of the func- tions a ± 21n (β ◦ ϑ−1 `∗ )(θ). In blue the set Γϕ(`∗ ,n,α,δ,a,a+ ,a− ) induc- tively defining the pseudo-curve. For every θ ∈ S1 and δ < 12 , ϑθ : [−δ, δ] −→ S1 denotes the map defined by ϑθ (x) = x + θ (mod 1). Clearly ϑθ is a homeomorphism between [−δ, δ] and Bδ [θ] . Finally ϑ−1 θ : Bδ [θ] −→ [−δ, δ] denotes the inverse homeomorphism of ϑθ . Definition 3.1 (Generic boxes). Fix `, n ∈ Z, n ≥ |`| , α ∈ (0, 2−n ), δ ∈ (0, α), a ∈ [−1, 1] and a+ , a− ∈ Ba(2−n β(δ)) (see Figure 3). Now we consider the Jordan closed curve in Ω, formed by the graphs of the functions a + 2−n (β ◦ ϑ−1 ∗ ) ∗ and a − 2−n (β ◦ ϑ−1 ∗ ) ∗ , ` Bδ [` ] ` Bδ [` ] together with the four segments that join the points:  (`∗ − α, a− ) with `∗ − δ, a − 2−n β(−δ) ,  (`∗ − α, a− ) with `∗ − δ, a + 2−n β(−δ) ,  (`∗ + α, a+ ) with `∗ + δ, a − 2−n β(δ) , and  (`∗ + α, a+ ) with `∗ + δ, a + 2−n β(δ) . We denote the closure of the connected component of the complement of the above Jordan curve in Ω that contains the point (`∗ , a) by R(`∗ , n, α, δ, a, a+ , a− ) (the coloured region in Figure 3). Observe that π (R(`∗ , n, α, δ, a, a+ , a− )) , the projec- tion of R(`∗ , n, α, δ, a, a+ , a− ) to S1 , is Bα[`∗ ] = [`∗ − α, `∗ + α]. We denote by ϕ`∗ = ϕ(`∗ ,n,α,δ,a,a+ ,a− ) : Bα[`∗ ] \{`∗ } −→ I the continuous map defined as follows: (i) ϕ`∗ B [`∗ ]\{`∗ } = a + (−1)` 2−n (φ ◦ ϑ−1 `∗ ). δ ∗ − ∗ + (ii) ϕ`∗ (` − α) = a and ϕ `∗ (` + α) = a . (iii) ϕ`∗ [`∗ −α,`∗ −δ] and ϕ`∗ [`∗ +δ,`∗ +α] are affine. 10 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES We also denote by Γϕ(`∗ ,n,α,δ,a,a+ ,a− ) ⊂ R(`∗ , n, α, δ, a, a+ , a− ) the closure in Ω of the graph of ϕ(`∗ ,n,α,δ,a,a+ ,a− ) . Remark 3.2. The region R(`∗ , n, α, δ, a, a+ , a− ) and the set Γϕ(`∗ ,n,α,δ,a,a+ ,a− ) sat- isfy the following properties: (1) R(`∗ , n, α, δ, a, a+ , a− ) ⊂ Bα[`∗ ] × [a − 2−n , a + 2−n ]. ∗ (2) diam(R(`∗ , n, α, δ, a, a+ , a− )) = diam(R(`∗ , n, α, δ, a, a+ , a− )` ) = 2 · 2−n . (3) The sets Γϕ(`∗ ,n,α,δ,a,a+ ,a− ) and ∂R(`∗ , n, α, δ, a, a+ , a− ) only intersect at the points (`∗ , a − 2−n ), (`∗ , a + 2−n ), (`∗ − α, a− ) and (`∗ + α, a+ ).   `∗ ∗ (4) Γϕ(`∗ ,n,α,δ,a,a+ ,a− ) = R(`∗ , n, α, δ, a, a+ , a− )` is an interval. (5) Let R(`∗ , n, α, δ, a, a+ , a− ) and R(k ∗ , n e, α ee e, δ, a− ) be two regions, then a+ , e a, e ∗ ∗ Bα[` ] ∩ Bαe[k ] = ∅ implies R(`∗ , n, α, δ, a, a+ , a− ) ∩ R(k ∗ , n e, α ee e, δ, a− ) = ∅. a+ , e a, e For every j ∈ Z+ , we set Zj := {i ∈ Z : |i| ≤ j} = {−j, −j + 1, . . . , −1, 0, 1, . . . , j − 1, j} and Zj∗ := {i∗ : i ∈ Zj }. With the help of the sets R(`∗ , n, α, δ, a, a+ , a− ) and Γϕ(`∗ ,n,α,δ,a,a+ ,a− ) , which are the “bricks” of our construction we are ready to define the sequence of pseudo-curve generators {(γj , S1 \Zj∗ )}∞ j=0 that we are looking for. To do this, for every j ≥ 0 we define j=0 ⊂ N, • a strictly increasing sequence {nj }∞ • a strictly decreasing sequence {αj }∞j=0 such that 2 −nj+1 < αj < 2−nj ∞ −nj+1 • and a sequence {δj }j=0 with 2 < δj < αj verifying some technical properties that we will make explicit below, and we de- − fine a sequence of boxes R(j ∗ ) := R(j ∗ , nj , αj , δj , aj , a+ ∗ j , aj ) and R((−j) ) := ∗ + − R((−j) , nj , αj , δj , a−j , a−j , a−j ) (for j = 0 both sets coincide) with projections π (R(j ∗ )) = Bαj [j ∗ ] and π (R((−j)∗ )) = Bαj [(−j)∗ ] . Finally, with the use of all these sequences and objects we can define our functions γj 1 ∗ . S \Zj Observe that we are using the intervals of the form Bα|`| [`∗ ] , Bδ|`| [`∗ ] and also Bα|`|−1 [`∗ ] when ` is negative. To ease the use of these intervals we introduce the following notation: ( ( ∗ Bα` [`∗ ] if ` ≥ 0, or ∗ Bα` (`∗ ) if ` ≥ 0, or B` [` ] := ∗ and B ` (` ) := ∗ Bα|`+1| [` ] if ` < 0, Bα|`+1| (` ) if ` < 0. Notice that the ball B` [`∗ ] has diameter αj for ` ∈ {j, −(j + 1)}. Remark 3.3. With the above notation Bα|`| [`∗ ] B` (`∗ ) for every ` < 0. More- over, for ` ∈ Z and j ∈ Z+ ,  Rω Bαj [`∗ ] = Bαj [(` + 1)∗ ] , and (  Bα` [(` + 1)∗ ] if ` ≥ 0, or Rω B` [`∗ ] = Bα|`+1| [(` + 1)∗ ] if ` < 0. Also, the same formulae holds with α replaced by δ and for open balls. A SKEW-PRODUCT WITHOUT FIXED-CURVES 11 The next crucial definition fixes in detail all quantities and objects mentioned above. Definition 3.4. We start by defining R(0∗ ) := R(0∗ , n0 , α0 , δ0 , 0, 0, 0) and ϕ0∗ := ϕ(0∗ ,n0 ,α0 ,δ0 ,0,0,0) by choosing (Definition 3.1) n0 = 1, α0 < 12 = 2−n0 and δ0 < α0 small enough so that the intervals B0 [0∗ ] = Bα0 [0∗ ] , Bα0 [1∗ ] and B−1[(−1)∗ ] = Bα0 [(−1)∗ ] are pairwise disjoint; and (−2)∗ , 2∗ ∈ / B−1[(−1)∗ ] and, additionally, ∗ ∗ Bd (Bα0 [0 ]) ∩ O (ω) = ∅. − We also set a+ 0 = a0 = a0 = 0, and we define the map γ0 : S \{0} −→ I by 1 ( ϕ0∗ (θ) if θ ∈ Bα0 [0∗ ] \{0}, γ0 (θ) = 0 / Bα0 [0∗ ]. if θ ∈ For consistency with the definition of γj in the case j ≥ 1, we define the map γ−1 : S1 \{0} −→ I by γ−1 (θ) = 0 for every θ ∈ S1 . Then, notice that, a0 = γ−1 (0∗ ), a± ∗ ∗ 0 = ϕ0∗ (0 ± α0 ) = γ−1 (0 ± α0 ), and γ0 (θ) = γ−1 (θ) for every θ ∈ / Bα0 [0∗ ] . Next, for every j ∈ N we define R(j ∗ ), R((−j)∗ ) and (γj , S1 \Zj∗ ) from the corresponding boxes R(i∗ ) and Bα|i| [i∗ ] ⊂ Bi [i∗ ] for i ∈ Zj−1 , and (γj−1 , S1 \Zj−1 ∗ ) as follows. We take nj , δj and αj such that (see Figure 4 to fix ideas): (R.1) nj > nj−1 , δj < αj < 2−nj < δj−1 < αj−1 and    Bd Bαj [(−j)∗ ] ∪ Bd Bαj [j ∗ ] ∩ O∗ (ω) = ∅. (R.2) The intervals Bj [j ∗ ] = Bαj [j ∗ ] ,  Rω Bαj [j ∗ ] = Bαj [(j + 1)∗ ] , B−j [(−j)∗ ] = Bαj−1 [(−j)∗ ] and B−(j+1) [(−(j + 1))∗ ] = Bαj [(−(j + 1))∗ ] are pairwise disjoint,    γj−1 Bαj [`∗ ] ⊂ γj−1 (`∗ ) − 2−nj , γj−1 (`∗ ) + 2−nj for every ` ∈ {j + 1, −(j + 1)}, B` [`∗ ] ∩ Zj+1 ∗ = {`∗ } for ` ∈ {j, −(j + 1)} and Bαj [(j + 1)∗ ] ∩ Zj+1 ∗ = {(j + 1)∗ }, and (−(j + 2))∗ , (j + 2)∗ ∈ / B−(j+1)[(−(j + 1))∗ ] = Bαj [(−(j + 1))∗ ] .   (R.3) Bd Bα|k| [(k + 1)∗ ] ∩ Bαj [j ∗ ] ∪ Bαj [(−j)∗ ] = ∅ for every k ∈ Zj−1 . (R.4) Assume that there exists k ∈ Zj−1 such that Bαj [(j + 1)∗ ] ∩ Bk [k ∗ ] 6= ∅ and |k| is maximal verifying these conditions. Then, Bαj [(j + 1)∗ ] is contained in one of the two connected components of Bα|k| (k ∗ ) \{k ∗ } when Bαj [(j + 1)∗ ]∩ Bα|k| [k ∗ ] 6= ∅, and Bαj [(j + 1)∗ ] is contained in one of the two connected components of Bk (k ∗ ) \Bα|k| [k ∗ ] if Bαj [(j + 1)∗ ] ∩ Bα|k| [k ∗ ] = ∅ (note that, in this case, k must be negative). (R.5) Let ` ∈ {j, −(j + 1)} (recall that the ball B` [`∗ ] has diameter αj for these two values of ` and S only for them). (R.5.i) If `∗ ∈ / i∈Zj−1 Bi [i∗ ] then, B` [`∗ ] ∩ Bi [i∗ ] = ∅ for every i ∈ Zj−1 . (R.5.ii) If `∗ ∈ Bm[m∗ ] for some m ∈ Zj−1 such that |m| is maximal with these properties, then (R.5.ii.1) B` [`∗ ] ∩ Bi [i∗ ] = ∅ for every i ∈ Zj−1 such that |i| ≥ |m| , i 6= m, and 12 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES (R.5.ii.2) B` [`∗ ] is contained in (a connected component of)   Bm (m∗ ) \ Bd Bα|m| [m∗ ] ∪ {m∗ } =   m∗ − α|m|−1 , m∗ − α|m| ∪ m∗ − α|m| , m∗ ∪   m∗ , m∗ + α|m| ∪ m∗ + α|m| , m∗ + α|m|−1 (observe that B` [`∗ ] ⊂ Bm(m∗ ) \Bα|m| [m∗ ] can only hap- pen when m < 0 since Bm[m∗ ] = Bα|m| [m∗ ] for m ≥ 0). (R.6) Let ` ∈ {j, −j}. If B` [`∗ ] ∩ Bm[m∗ ] = ∅ for every m ∈ Zj , m 6= ` then, to define R(`∗ ) and the map ϕ`∗ , we set a` = γj−1 (`∗ ) = a± ∗ ` = γj−1 (` ± αj ) = 0. ∗ Otherwise, there exists m ∈ Zj−1 such that  B` [`∗ ]is contained in a connected ∗ ∗ component of Bm(m ) \ Bd Bα|m| [m ] ∪ {m } and |m| is maximal with these properties. Then, to define R(`∗ ) and the map ϕ`∗ , we set  (R.6.i) a` := γ (`∗ ), a± := γ (`∗ ±αj ) and Graph γ |m| ` |m| |m| ∗ ⊂ R(`∗ ). Bαj [` ] (R.6.ii) Assume that there exists k ∈ Z|m| ⊂ Zj−1 such that B` [`∗ ] ⊂ Bα|k| (k ∗ ) \{k ∗ }. Then, R(`∗ ) is contained in one of the two con-   nected components of Int R(k ∗ )\  k ∗ . Finally we define γj : S1 \Zj∗ −→ I by   ϕj∗ (θ) if θ ∈ Bαj [j ∗ ] \{j ∗ }, γj (θ) = ϕ(−j)∗ (θ) if θ ∈ Bαj [(−j)∗ ] \{(−j)∗ },    γj−1 (θ) / Bαj [j ∗ ] ∪ Bαj [(−j)∗ ] ∪ Zj−1 if θ ∈ ∗ . (notice that Zj∗ = Zj−1 ∗ ∪ {j ∗ , (−j)∗ }). For every ` ∈ Z we define the winged region associated to ` as  R(`∗ ) R (`∗ ) :=   if ` ≥ 0, or ∗ R(` ) ∪ Graph γ|`| B [`∗ ]\B (`∗ ) if ` < 0. ` α|`| The next technical lemma shows that the objects from Definition 3.4 exist (that is, they are well defined), and studies some of the basic properties of the family of pseudo-curve generators {(γi , S1 \Zi∗ )}∞ i=0 . Remark 3.5 (Explicit consequences of Definition 3.4). The following statements are easy consequences of Definition 3.4. They are stated explicitly for easiness of usage. (R.1) nj > j. This follows from Definition 3.4(R.1) and the fact that we have set n0 = 1 and nj > nj−1 for j ∈ N. (R.2) For every j ∈ N, B−j [(−j)∗ ] ∩ Zj+1 ∗ = {(−j)∗ }. This follows from Definition 3.4(R.2) for j − 1. We get B−j [(−j)∗ ] ∩ Zj∗ = {(−j)∗ } and (−(j + 1))∗ , (j + 1)∗ ∈ / B−j [(−j)∗ ] . which shows the statement. (R.6) Let j ∈ N and ` ∈ {j, −j}, and assume ∗ ∗ that B` [` ] ∩ Bm[m ] = ∅ for every m ∈ Zj , m 6= `. Then, γr B [`∗ ] = γ0 B [`∗ ] ≡ 0 for r = 1, 2, . . . , j − 1. ` ` (R.6.i) Assume that here exists m ∈ Zj−1 such that B` [`∗ ] is  contained  in a connected component of Bm(m∗ ) \ Bd Bα|m| [m∗ ] ∪ {m∗ } and A SKEW-PRODUCT WITHOUT FIXED-CURVES 13 R((−3)∗ ) R((−4)∗ ) R(3∗ ) R(4∗ ) R((−2)∗ ) R(2∗ ) R(1∗ ) R((−1)∗ ) R(0∗ ) Figure 4. The boxes R(`∗ ) for ` ∈ {−4, −3, −2, −1, 0, 1, 2, 3, 4} and the graph of γ4 . The wings are represented as a thick garnet curve surrounding the graph of γ4 . For clarity the scale and separa- tion between boxes is not preserved. The circle S1 is parametrized as [− 21 , 12 ). |m| is maximal with these properties. Then, γr B [`∗ ] = γ|m| B [`∗ ] ` ` for r = |m| + 1, |m| + 2, . . . , j − 1. (R.6.ii) Assume that there exists k ∈ Z|m| ⊂ Zj−1 such that B` [`∗ ] ⊂ Bα|k| (k ∗ ) \{k ∗ } and |k| is maximal with these properties. Then, γr B [`∗ ] = γ|k| B [`∗ ] for r = |k| + 1, |k| + 2, . . . , |m| . ` ` To prove (R.6) notice that when B` [`∗ ]∩Bα|m| [m∗ ] ⊂ B` [`∗ ]∩Bm[m∗ ] = ∅ for every m ∈ Zj , m 6= `, from the definition of γr for 0 ≤ r < j we get that γr B [`∗ ] = ` γ0 ∗ ≡ 0 for r = 1, 2, . . . , j − 1. B` [` ] (R.6.i) The maximality of |m| , together with Definition 3.4(R.2), imply that B` [`∗ ] ∩ Bα|i| [i∗ ] ⊂ B` [`∗ ] ∩ Bi [i∗ ] = ∅ for every i ∈ Zj−1 , |i| ≥ |m| , i 6= m. So, by the definition of the functions γr , γr Bα [`∗ ] = γ|m| Bα [`∗ ] for r = |m| + 1, |m| + 2, . . . , j − 1. j j (R.6.ii) When |k| = |m| (R.6.ii) holds trivially. So, assume that |k| < |m| . As in the case (R.6.i), the maximality of |k| and Definition 3.4(R.2) imply that B` [`∗ ] ∩ Bα|r| [r∗ ] = ∅ for every r ∈ Zj−1 , |r| ≥ |k| , r 6= k. So, (R.6.ii) follows from the definition of the functions γr . Lemma 3.6. For every j ∈ Z+ the regions R(j ∗ ) and R((−j)∗ ) (and hence R (j ∗ ) and R ((−j)∗ )), and the maps (γj , S1 \Zj∗ ) are well defined. Moreover, the following statements hold: (a) (γj , S1 \Zj∗ ) ∈ PCG. Furthermore, for every ` ∈ {j + 1, −(j + 1)},    γj Bαj [`∗ ] ⊂ γj (`∗ ) − 2−nj , γj (`∗ ) + 2−nj . 14 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES S   (b) R (`∗ ) ⊂ S1 × [−1, 1] and Graph γj S1 \Z ∗ ⊂ S1 × [−1, 1]. `∈Z  j (c) For ` ∈ {j, −j} we have Graph γj−1 B [`∗ ] ⊂ R(`∗ ), a` = γj−1 (`∗ ), and αj a± ` =ϕ`∗ (`∗ ± αj ) =γj−1 (`∗ ± αj ). (d) Graph γn B [`∗ ]\Z ∗ ⊂ R(`∗ ) for every n ≥ j and ` ∈ {j, −j}. αj n (e) For every ` ∈ {j, −j}, γj (B [`∗ ]\B (`∗ ))∪R (B [`∗ ]\B (`∗ ) ) = γ j−1 (B` [`∗ ]\Bαj (`∗ ))∪Rω (B` [`∗ ]\Bαj (`∗ )) . ` αj ω ` αj Moreover, for every θ ∈ Bd(B` [`∗ ] \Bαj (`∗ )) = Bd(Bαj [`∗ ]) ∪ Bd(B` [`∗ ]), we have θ ∈ / Bn [n∗ ] ∪ B−n[(−n)∗ ] and γn (θ) = γj (θ) = γj−1 (θ) for every  n > j, and Rω (θ) ∈ / Bαn [n∗ ] ∪ Bαn [(−n)∗ ] and γn Rω (θ) = γj−1 Rω (θ) for every n ≥ j.  (f ) For every ` ∈ Z, R (`∗ ) is a compact connected set such that π R (`∗ ) = B` [`∗ ] , γ|`| B [`∗ ]\Bα (`∗ ) is continuous and ` |`| (  diam (R(`∗ )) = diam (R((−`)∗ )) = 2 · 2−n` ≤ 2−` if ` ≥ 0, diam R (`∗ ) = 2 · 2−n|`+1| ≤ 2 · 2−|`| if ` < 0. (g) Given `, m ∈ Z such that |`| ≥ |m| , ` 6= m and B` [`∗ ] ∩ Bm[m∗ ] 6= ∅, it follows that |`| > |m| , and either B` [`∗ ] ⊂ Bα|m| (m∗ ) \{m∗ } and the region R (`∗ )   is contained in one of the two connected components of Int R(m∗ )\  m∗ , or m < 0 and B` [`∗ ] is contained in one of the two connected components of Bm(m∗ ) \Bα|m| [m∗ ]. Proof. We start by proving the first statement of the lemma and (a) by induction. Observe that n0 = 1, α0 , δ0 and γ0 are defined so that Definition 3.4(R.1–2) for j = 0 and (γ0 , S1 \Z0∗ ) ∈ PCG are verified except for the obvious fact that B−j [(−j)∗ ] = Bj [j ∗ ] . On the other hand, by construction, Bα0 [0∗ ] is disjoint from Bα0 [1∗ ] and Bα0 [(−1)∗ ] . Then, by the definition of γ0 ,   γ0 (Bα0 [`∗ ]) = {0} ⊂ [− 21 , 12 ] = γ0 (`∗ ) − 2−n0 , γ0 (`∗ ) + 2−n0 for ` ∈ {1, −1}. Hence, (a) holds. Fix j > 0 and assume that we have defined n` , α` , δ` and γ` such that all Definition 3.4(R.1–6) above and (a) hold for ` = 0, 1, . . . , j − 1. ∗ Since the elements of Zj+2 are pairwise different, we can choose an integer nj > nj−1 and δj and αj small enough so that • 0 < δj < αj < 2−nj < δj−1 , • (−(j + 2))∗ , (j + 2)∗ ∈ / B−(j+1)[(−(j + 1))∗ ] = Bαj [(−(j + 1))∗ ] ,  • the three intervals Bj [j ∗ ] = Bαj [j ∗ ] , Rω Bαj [j ∗ ] = Bαj [(j + 1)∗ ] and B−(j+1)[(−(j + 1))∗ ] are pairwise disjoint, • B` [`∗ ] ∩ Zj+1 ∗ = {`∗ } for ` ∈ {j, −(j + 1)}, ∗ ∗ Bαj [(j + 1) ] ∩ Zj+1 = {(j + 1)∗ } and, additionally,    • Bd Bαj [(−j)∗ ] ∪ Bd Bαj [j ∗ ] ∩ O∗ (ω) = ∅. Then, Definition 3.4(R.1) is verified. Moreover, from the above conditions it follows that Bαj [`∗ ] ∩ Zj+1 ∗ = {`∗ } for every ` ∈ {j + 1, −(j + 1)}. Thus, by statement (a) for j − 1, γj−1 is defined and continuous on `∗ ∈ Bαj [`∗ ] because this interval is ∗ disjoint from Zj−1 . Hence, we can decrease the value of αj (and, accordingly, the value of 0 < δj < αj ), if necessary, to get A SKEW-PRODUCT WITHOUT FIXED-CURVES 15    • γj−1 Bαj [`∗ ] ⊂ γj−1 (`∗ ) − 2−nj , γj−1 (`∗ ) + 2−nj for every ` ∈ {j + 1, −(j + 1)}. To see that Definition 3.4(R.2) is verified it remains to show that the intervals Bj [j ∗ ] , Bαj [(j + 1)∗ ] and B−(j+1)[(−(j + 1))∗ ] are disjoint from B−j [(−j)∗ ] . By in- duction, Definition 3.4(R.2) holds for j−1. Thus we see, that (−(j + 1))∗ , (j + 1)∗ ∈ / B−j [(−j)∗ ] , and Rω Bαj−1 [(j − 1)∗ ] = Bαj−1 [j ∗ ] is disjoint from B−j [(−j)∗ ] . Hence, we can decrease the value of αj (and, accordingly, the value of 0 < δj < αj ), if necessary, until Bαj [(j + 1)∗ ] and B−(j+1)[(−(j + 1))∗ ] = Bαj [(−(j + 1))∗ ] are disjoint from B−j [(−j)∗ ] . On the other hand we have that αj < 2−nj < δj−1 < αj−1 . So, Bj [j ∗ ] = Bαj [j ∗ ] ⊂ Bαj−1 [j ∗ ] is disjoint from B−j [(−j)∗ ] . Up to now we have seen that we can choose nj , δj and αj so that Defini- tion 3.4(R.1–2) hold for j. Let us see that we can choose αj such that Defini- tion 3.4(R.3) also holds. Observe that for every `, i ∈ Z and every m ≥ 0 it follows that Bd (Bαm [`∗ ]) ∩ O∗ (ω) 6= ∅ if and only if Bd Rωi (Bαm [`∗ ]) ∩ O∗ (ω) = Bd (Bαm [(` + i)∗ ]) ∩ O∗ (ω) 6= ∅. Therefore, by using Definition 3.4(R.1) inductively, we obtain [  [  Bd Bα|k| [(k + 1)∗ ] ∩{(−j)∗ , j ∗ } ⊂ Bd Bα|k| [(k + 1)∗ ] ∩O∗ (ω) = ∅. k∈Zj−1 k∈Zj−1 S  Consequently, since k∈Zj−1 Bd Bα|k| [(k + 1)∗ ] is a finite set, by decreasing again the value of αj , if necessary, we can achieve that Definition 3.4(R.3) holds for j and Definition 3.4(R.1–2) are still verified. S Next we will take care of Definition 3.4(R.4). If (j + 1)∗ ∈ / i∈Zj−1 Bi [i∗ ] , by decreasing again S the value  of αj (and δj ), if necessary, we can achieve that ∗ ∗ Bαj [(j + 1) ] ∩ i∈Zj−1 Bi [i ] = ∅ while preserving that Definition 3.4(R.1–3) are verified for j. In this case Definition 3.4(R.4) holds trivially. Conversely, assume that there exists k ∈ Zj−1 such that (j + 1)∗ ∈ Bk [k ∗ ] and |k| is maximal verifying these conditions. By Definition 3.4(R.2), k is unique (that is, the condition cannot be verified by k and −k simultaneously). On the other hand, by the Definition 3.4(R.1) for |k| and |k| − 1 and the comment above, (j + 1)∗ ∈/ Bd (Bk [k ∗ ]) ∪ Bd Bα|k| [k ∗ ] . Since k ∈ Zj−1 , |k| ≤ j − 1 and, hence, ∗ ∗ (j + 1) ∈/ Z|k| (in particular j ∗ 6= k ∗ ). Consequently, (j + 1)∗ is contained in one of    the connected components of Bk (k ∗ ) \ Bd Bα|k| [k ∗ ] ∪ Z|k| ∗ . Then, by decreas- ing again the value of αj , if necessary, we  can get that Bαj [(j + 1)∗ ] is contained  in the connected component of Bk (k ∗ ) \ Bd Bα|k| [k ∗ ] ∪ Z|k| ∗ where (j + 1)∗ lies, while preserving that Definition 3.4(R.1–3) are verified for j. Consequently, Defini- tion 3.4(R.1–4) hold for j. S Now we will deal with Definition 3.4(R.5). If `∗ ∈/ i∈Zj−1 Bi [i∗ ] , by decreasing again the value of αj , if necessary, we can get Definition 3.4(R.5.i) while preserving that Definition 3.4(R.1–4) are verified for j. Assume that there exists m ∈ Zj−1 such that `∗ ∈ Bm[m∗ ] and |m| is maxi- mal with these properties. As in  the above  construction, by Definition 3.4(R.1–2), `∗ ∈ Bm(m∗ ) \ Bd Bα|m| [m∗ ] ∪ {m∗ } and m is unique (that is, the condition cannot be verified simultaneously by m and −m). Consequently, `∗ ∈ / Bi [i∗ ] for every i ∈ Zj−1 such that |i| ≥ |m| , i 6= m. Thus, by decreasing again the value of αj , if necessary, we can get that Definition 3.4(R.1–4) still hold, Definition 3.4(R.5.ii.1) ∗ is verified and the interval B` ∗[` ] is contained in the connected component of Bm(m ) \ Bd Bα|m| [m ] ∪ {m } where `∗ lies. So, Definition 3.4(R.5.ii.2) also ∗ ∗ holds. 16 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES We claim that for every `, m ∈ Z such that |m| ≤ |`| ≤ j, ` 6= m, either B` [`∗ ] ∩ Bm[m∗ ] = ∅ or |m| < |`| and B` [`∗ ] is contained in a connected component of   Bm (m∗ ) \ Bd Bα|m| [m∗ ] ∪ {m∗ } . We prove the claim by induction. Observe that the claim holds trivially for |m| ≤ |`| ≤ 1 because B0 [0∗ ] , B1 [1∗ ] = Bα1 [1∗ ] ⊂ Bα0 [1∗ ] and B−1[(−1)∗ ] are pairwise disjoint by construction. Assume that the claim holds for every |m| ≤ |`| < j. So, to prove the claim, we may assume that ` ∈ {j, −j}, m ∈ Zj−1 ∪ {−`} and B` [`∗ ] ∩ Bm[m∗ ] 6= ∅. By Definition 3.4(R.2), Bj [j ∗ ] ∩ B−j [(−j)∗ ] = ∅. Consequently, m 6= −` (that is, m ∈ Zj−1 and |`| = j > |m|). On the other hand, if ` = −j, Definition 3.4(R.2) for j − 1 shows that Bj−1[(j − 1)∗ ] , B−(j−1)[(−(j − 1))∗ ] and B−j [(−j)∗ ] are pairwise disjoint. Thus, m ∈ Zj−2 in this case. Hence, by the Definition 3.4(R.5) for j when ` = j and for j−1 when ` = −j, there ∗ exists k ∈ Zj−1 (in fact when ` = −j, k ∈ Zj−2 )such that  B` [` ] is contained in a ∗ ∗ ∗ connected component of Bk (k ) \ Bd Bα|k| [k ] ∪ {k } and |`| = j > |k| ≥ |m| . If m = k then the claim holds. Otherwise, m 6= k and since j = |`| > |k| ≥ |m|, by the induction hypotheses, |k| > |m|, and Bk [k ∗ ] is contained in a connected component of Bm(m∗ ) \ Bd Bα|m| [m∗ ] ∪ {m∗ } . So, the claim holds also in this case. This ends the proof of the claim. Finally, we consider Definition 3.4(R.6). The fact that either B` [`∗ ]∩Bm[m∗ ] = ∅ ∗ for every m ∈ Zj , m 6= ` or there exists m ∈ Zj−1 such  that∗ B` [` ] is contained ∗ ∗ in a connected component of Bm(m ) \ Bd Bα|m| [m ] ∪ {m } follows from the claim. To show that Definition 3.4(R.6.i) can be guaranteed, it is enough to decrease again the value of αj , if necessary, until Bαj [`∗ ] is disjoint from Z|m| ∗ and Defi- nition 3.4(R.1–5) are still verified. Thus by (a) for |m| , γ|m| is well defined and continuous on Bαj [`∗ ] . So, we can set a` := γ|m| (`∗ ) and, by decreasing again αj   (if necessary), we get Graph γ |m| ∗ ⊂ R(j ∗ ). Bαj [` ] To show that Definition 3.4(R.6.ii) can be guaranteed we first assume that k = m. As before, if necessary, we can increase the value of nj and, accordingly, decrease the values of αj < 2−nj and 0 < δj < αj so that Definition 3.4(R.1–5) and (R.6.i) are still verified for j and in addition, (`∗ , a` + 2−nj ), (`∗ , a` − 2−nj ) ∈ Int(R(k ∗ )) and the region  R(`∗ ) is contained in one of the two connected components of Int R(k ∗ )\  k ∗ . Assume now that k 6= m (recall that |k| ≤ |m| < j). In this case we have B` [`∗ ] ⊂ Bm(m∗ ) ∩ Bα|k| (k ∗ ). In particular, Bm(m∗ ) ∩ Bα|k| (k ∗ ) 6= ∅ and, by the above claim, |k| < |m| and B` [`∗ ] ⊂ Bm[m∗ ] is contained in a connected component of Bk (k ∗ ) \ Bd Bα|k| [k ∗ ] ∪ {k ∗ } . The fact that B` [`∗ ] ⊂ Bα|k| (k ∗ ) \{k ∗ } implies that B` [`∗ ] ⊂ Bm[m∗ ] ⊂ Bα|k| (k ∗ ) \{k ∗ }. Then, as above we can increase the value of nj and, accordingly, decrease the values of αj < 2−nj and 0 < δj < αj so that Definition 3.4(R.1–5) and (R.6.i) are still verified, (`∗ , a` + 2−nj ), (`∗ , a` − 2−nj ) ∈ Int(R(k ∗ )) and the region  R(`∗ ) is contained in one of the two connected components of Int R(k ∗ )\  k ∗ . A SKEW-PRODUCT WITHOUT FIXED-CURVES 17 Now assume that |k| is not maximal verifying the assumptions. Then, there exists r ∈ Z|m| ⊂ Zj−1 such that B` [`∗ ] ⊂ Bα|r| (r∗ ) \{r∗ } and |r| is maximal with these properties. We have |k| ≤ |r| ≤ |m| < j and Br [r∗ ] ∩ Bk [k ∗ ] ⊃ Bα|r| (r∗ ) ∩ Bα|k| (k ∗ ) 6= ∅ because B` [`∗ ] ⊂ Bα|r| (r∗ ) ∩ Bα|k| (k ∗ ) . Then, by the claim, |k| < |r| and Br [r∗ ] is   contained in a connected component of Bk (k ∗ ) \ Bd Bα|k| [k ∗ ] ∪ {k ∗ } . The fact that B` [`∗ ] ⊂ Bα|k| (k ∗ ) \{k ∗ } implies that Br [r∗ ] ⊂ Bα|k| (k ∗ ) \{k ∗ }. By the part already proven and Definition 3.4(R.6.ii) for |r|  < j we get that R(`∗ ) is contained in one of the two connected components of Int R(r∗ )\  r∗ and R(r∗ ) is contained   in one of the two connected components of Int R(k ∗ )\  k ∗ . This shows that Definition 3.4(R.6.ii) can be guaranteed. set S \Zj is residual, to prove that 1 ∗ Let us prove that (a) holds for j. Since the (γj , S \Zj ) ∈ PCG we have to show that γj S1 \Z ∗ is continuous. Note that, from 1 ∗ j Definition 3.4(R.6.ii), a± ∗ ∗ ` = ϕ`∗ (` ± αj ) = γj−1 (` ± αj ). Hence, the continuity of γj S1 \Z ∗ follows from the fact that γj−1 is continuous on S1 \Zj−1 ∗ ⊃ S1 \Zj∗ and the j continuity of ϕj∗ and ϕ(−j)∗ (Definition 3.1). This ends the proof of the first statement of the lemma and the first statement of (a). For every ` ∈ {j + 1, −(j + 1)}, from By Definition 3.4(R.1,2) we get:    γj−1 Bαj [`∗ ] ⊂ γj−1 (`∗ ) − 2−nj , γj−1 (`∗ ) + 2−nj Bαj [`∗ ] is disjoint from Bαj [j ∗ ] and Bαj−1 [(−j)∗ ] ⊃ Bαj [(−j))∗ ], and {`∗ } ∈ / Bαj [`∗ ] ∩ Zj−1 ∗ ⊂ Bαj [`∗ ] ∩ Zj+1 ∗ = {`∗ }. So, from the definition of γj it follows that γj B [`∗ ] = γj−1 B ∗] αj αj [` and, thus, (a) holds. Statement (c) follows immediately from Definition 3.4(R.6) and Remark 3.5(R.6). Next we prove (b,d,e,f,g). (d) When n = j, we get Bαj [`∗ ] \Zj∗ = Bαj [`∗ ] \{`∗ } from Definition 3.4(R.2).   Hence, Graph γj B [`∗ ]\Z ∗ ⊂ R(`∗ ) by the definition of γj (Definition 3.4) and αj j the definition of ϕ`∗ (Definition 3.1). Now assume that n > j and fix θ ∈ Bαj [`∗ ] \Zn∗ . We have to show that the point (θ, γn (θ)) ∈ R(`∗ ). If θ ∈ / Bα|m| [m∗ ] for every m such that j < |m| ≤ n then, by the iterative use of the definition of γi for i = j + 1, j + 2, . . . , n (Definition 3.4) and Definition 3.1, (θ, γn (θ)) = (θ, γn−1 (θ)) = · · · = (θ, γj+1 (θ)) = (θ, γj (θ)) = (θ, ϕ`∗ (θ)) ∈ R(`∗ ). Otherwise, by Definition 3.4(R.2), there exists m ∈ Z such that |`| < |m| ≤ n, θ ∈ Bα|m| [m∗ ] \Zn∗ , and θ ∈ / Bα|s| [s∗ ] for every s such that |m| < |s| ≤ n. This im- plies that B` [`∗ ]∩Bm[m∗ ] ⊃ Bαj [`∗ ]∩Bα|m| [m∗ ] 6= ∅ and |m| is maximal with these properties. So, by the claim for j = |m| , Bm[m∗ ] is contained in a connected compo- nent of B` (`∗ ) \ Bd Bα|`| [`∗ ] ∪ {`∗ } . Moreover, since θ ∈ Bm(m∗ ) ∩ Bαj [`∗ ] 6= ∅, Bm[m∗ ] ⊂ Bα|`| [`∗ ] \{`∗ }. Thus, by Definition 3.4(R.6.ii) and Remark 3.5(R.6.ii) for j = |m| , ` replaced by m and k replaced by `, R(m∗ ) ⊂ R(`∗ ) and (d) follows from the part already proven by replacing ` by m and j by |m| . 18 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES (g) By the claim we have that for every `, m ∈ Z such that |`| ≥ |m| , ` 6= m and B` [`∗ ] ∩ Bm[m∗ ] 6= ∅, it follows that |`| > |m| , and B` [`∗ ] is contained in a connected component of Bm(m∗ ) \ Bd Bα|m| [m∗ ] ∪ {m∗ } . Only it remains to show that if B` [`∗ ] ⊂ Bα|m| (m∗ ) \{m∗ }, then the region R (`∗ ) is contained in one   of the two connected components of Int R(m∗ )\  m∗ . By Definition 3.4(R.6.ii) we know that this holds for R(`∗ ) instead of R (`∗ ). Hence, if ` ≥ 0, (g) holds ∗ ∗ ∗ ∗  R (` ) = R(`  because ). Assume now that ` < 0. Since R (`  ) = R(` ) ∪  Graph γ |`|∗ ∗ is connected, R(` ) ⊂ R(m ), and Int R(m )\  m∗ B` [` ]\Bα|`| (` ) ∗ ∗ ∗ has two connected components, it is enough to show that   Graph γ|`| B [`∗ ]\B (`∗ ) ⊂ R(m∗ ). ` α|`| Since B` [`∗ ] \Bα|`| (`∗ ) ⊂ B` [`∗ ] ⊂ Bα|m| (m∗ ) \{m∗ }, statement (g) follows from (d) with ` replaced by m, j by |m| and n replaced by |`| . (b) With (g) in mind we set D := {` ∈ Z : R (`∗ ) 6⊂ R(i∗ ) for every i ∈ Z\{`}}. Clearly,   ! [ [ [ R (`∗ ) =  R (i∗ ) ∪ R (`∗ ) `∈Z i∈Z\D `∈D ! ! [ [ [ ∗ ∗ ⊂ R(i ) ∪ R (` ) = R (`∗ ) i∈D `∈D `∈D Claim: For every ` ∈ D, γ|`|−1 ≡ 0. B` [`∗ ]\Bα|`| (`∗ ) First we prove statement (b) from the above claim and then we will prove the claim. To this end we start by pointing out few elementary facts. ∗ ∗ ∗  R (` ) we see that From the definition of  R (` )\R(` ) = ∅ for every ` ≥ 0 and R (`∗ )\R(`∗ ) ⊂ Graph γ ∗ |`| ∗ for every ` < 0. So, in any case, B` [` ]\Bα|`| (` )   R (`∗ )\R(`∗ ) ⊂ Graph γ|`| B ∗ ∗ for every ` ∈ Z. ` [` ]\Bα|`| (` ) On the other hand, the arc B` [`∗ ] ⊃ B` [`∗ ] \Bα|`| (`∗ ) is disjoint from the arc B−`[(−`)∗ ] ⊃ Bα|`| [(−`)∗ ] by Definition 3.4(R.2). Thus, by Definition 3.4 and (a), γ|`|−1 B [`∗ ]\Bα (`∗ ) = γ|`| B [`∗ ]\Bα (`∗ ) . ` |`| ` |`| Furthermore, by the Claim and Definition 3.4(R.6), a+ ` = a− ` = a` = 0 for every ` ∈ D. So, by Remark 3.2(1), R(`∗ ) ⊂ Bα|`| [`∗ ]×[−2−n|`| , 2−n|`| ] ⊂ Bα|`| [`∗ ]×[−2−|`| , 2−|`| ] ⊂ Bα|`| [`∗ ]×[−1, 1]. Therefore, summarizing and using again by the Claim, [ [ [   R (`∗ ) ⊂ R (`∗ ) ⊂ R(`∗ ) ∪ Graph γ|`| B [`∗ ]\Bα|`| (`∗ ) ` `∈Z `∈D `∈D ! ! [ [   = R(` ) ∗ ∪ Graph γ|`|−1 B [`∗ ]\Bα|`| (`∗ ) ` `∈D `∈D ! [ ∗ ⊂ Bα|`| [` ] × [−1, 1] ∪ S1 × {0} ⊂ S1 × [−1, 1]. `∈D A SKEW-PRODUCT WITHOUT FIXED-CURVES 19 So, the first part of (b) is proved, provided that the claim holds. Let us prove the second statement of (b). Observe that, since ! ! [ [ R(`∗ ) ∪ S1 × {0} ⊂ R (`∗ ) ∪ S1 × {0} ⊂ S1 × [−1, 1], `∈Z `∈Z it is enough to show that !   [ Graph γ j S1 \Zj∗ ⊂ ∗ R(` ) ∪ S1 × {0} `∈Z for every j ∈ Z . We will prove this statement by induction on j. + By construction we have !   [ Graph γ0 S1 \{0∗ } ⊂ R(0 ) ∪ S × {0} ⊂ ∗ 1 ∗ R(` ) ∪ S1 × {0}. `∈Z So, the statement holds for j = 0. Now assume that it holds for some j ≥ 0, and prove it for j + 1. By Definition 3.4 and (d),     Graph γj+1 S1 \Z ∗ ⊂ R(j ∗ ) ∪ R((−j)∗ ) ∪ Graph γj S1 \Z ∗ j+1 j ! [ ⊂ R(j ∗ ) ∪ R((−j)∗ ) ∪ R(`∗ ) ∪ S1 × {0} `∈Z ! [ ⊂ R(`∗ ) ∪ S1 × {0}. `∈Z To end the proof of (b) it remains to show the Claim. Let ` ∈ D and m ∈ Z|`| , m 6= `. Then, either ( B` [`∗ ] ∩ Bm[m∗ ] = ∅ or (1) |`| > |m| , m < 0 and B` [`∗ ] ⊂ Bm(m∗ ) \Bα|m| [m∗ ] . To see this, observe that if B` [`∗ ] ∩ Bm[m∗ ] 6= ∅ then, by (g), |`| > |m| and either R (`∗ ) ⊂ R(m∗ ) or m < 0 and B` [`∗ ] ⊂ Bm(m∗ ) \Bα|m| [m∗ ] , and the first possibility is ruled out because ` ∈ D. By using iteratively the dichotomy (1) we get that, for every ` ∈ D, there exists a sequence m0 , m1 , . . . , mk = ` ∈ Z with k ≥ 0 such that Bm0 [(m0 )∗ ] ∩ Bq [q ∗ ] = ∅ for every q ∈ Z|m0 | , q 6= m0 and, in the case k > 0, |m0 | < |m1 | < · · · < |mk | = |`| and, for every p = 0, 1, . . . , k − 1, • mp < 0, • Bmp+1 [(mp+1 )∗ ] ⊂ Bmp ((mp )∗ ) \Bα|mp | [(mp )∗ ] and • Bmp+1 [(mp+1 )∗ ] ∩ Bq [q ∗ ] = ∅ for every q ∈ Z|mp+1 | , q 6= mp , mp+1 and |mp | ≤ |q| . The condition Bm0 [(m0 )∗ ] ∩ Bq [q ∗ ] = ∅ for every q ∈ Z|m0 | , q 6= m0 implies γ|m0 |−1 B [(m )∗ ] = γ|m0 |−2 B [(m )∗ ] = · · · = γ0 B [(m )∗ ] ≡ 0 m0 0 m0 0 m0 0 by Definition 3.4(R.6) and Remark 3.5(R.6) (with ` = m0 ). This ends the proof of the Claim when k = 0. Assume now that k > 0. As before we have γ|m0 |−1 B [(m )∗ ]\B ((m0 )∗ ) = γ|m0 | B [(m0 )∗ ]\Bα ((m0 )∗ ) . m0 0 α |m0 | m0 |m0 | This, together with the inclusion, Bm1 [(m1 )∗ ] ⊂ Bm0 ((m0 )∗ ) \Bα|m0 | [(m0 )∗ ] 20 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES implies that γ|m0 | B ∗ ≡ 0. m1 [(m1 ) ] Then, by Definition 3.4(R.6.i) and Remark 3.5(R.6.i) with ` = m1 , 0 ≡ γ|m0 | B [(m )∗ ] = γ|m0 |+1 B [(m )∗ ] = · · · = γ|m1 |−1 B ∗] . 1 m1 1 m1 m1 [(m1 ) If k = 1 we are done. Otherwise, k ≥ 2 and, as above, γ|m1 | B [(m )∗ ] ≡ 0. m2 2 By iterating the above arguments at most k times the Claim holds. This ends the proof of (b). (e) By Definition 3.4(R.2) and Remark 3.5(R.2) it follows that ∗ θ∈ / Zj+1 ∪ Bαj (`∗ ) ∪ B−` ((−`)∗ ) for every θ ∈ B` [`∗ ] \Bαj (`∗ ) . So, by (a), γj−1 (θ) is well defined and γj−1 is continuous at θ. Thus, by the definition of γj (Definition 3.4) and the continuity of γj−1 at θ, γj (θ) = γj−1 (θ). Now assume that θ ∈ Bd(B` [`∗ ] \Bαj (`∗ )) = Bd(Bαj [`∗ ]) ∪ Bd(B` [`∗ ]). By (g), θ∈/ Bn [n∗ ] ∪ B−n[(−n)∗ ] for every n > j. So, by the iterative use of the definition of γi for i = j + 1, j + 2, . . . , n (Definition 3.4) we get γj (θ) = γj+1 (θ) = · · · = γn−1 (θ) = γn (θ). Now we prove the part of (e) concerning Rω (B` [`∗ ] \Bαj (`∗ )). We first assume that ` = j ≥ 0. Then, Bj [j ∗ ] = Bαj [j ∗ ] , θ ∈ Bd(Bαj [j ∗ ]) and Rω (θ) ∈ Bd(Bαj [(j + 1)∗ ]). ∗ Again by Definition 3.4(R.2), Rω (θ) ∈ / Zj+1 ∪ Bαj [j ∗ ] ∪ B−j [(−j)∗ ] . So, by (a) and   the definition  of γ j (Definition 3.4), γ j−1 R ω (θ) is well defined and γ j R ω (θ) = ∗ γj−1 Rω (θ) . By Definition 3.4(R.3) (with j = n and k =  ` = j), Rω (θ) ∈ / Bαn [n ] ∪ Bαn [(−n)∗ ] for every n > j. So, γn Rω (θ) = γj Rω (θ) as above. Assume now that ` = −j < 0. In this case we have B` [`∗ ] = Bα|`+1| [`∗ ] and, hence, Rω (θ) ∈ Bα|`+1| [(` + 1)∗ ] \Bαj ((` + 1)∗ ) . By Definition 3.4(R.1) we have Bαj [(` + 1)∗ ] ⊂ Bα|`+1| [(` + 1)∗ ] ⊂ B`+1 [(` + 1)∗ ] . Thus, Rω (θ) ∈ B`+1[(` + 1)∗ ] \{(` + 1)∗ }. Again by Definition 3.4(R.2) and Re- mark 3.5(R.2) (with j replaced by −(` + 1)), Rω (θ) ∈/ Z`∗ ∪ Bα−(`+1) [(−`)∗ ] ∪ B` [`∗ ] ⊃ Zj∗ ∪ Bαj [j ∗ ] ∪ B−j [(−j)∗ ] .  So, by (a) and the definition  of γj (Definition 3.4), γj−1 Rω (θ) is well defined and γj Rω (θ) = γj−1 Rω (θ) . ∗ ∗ To end the proof of (e), assume as above that θ ∈ Bd(B  αj [` ]) ∪ Bd(B` [` ]) and, hence, Rω (θ) ∈ Bd(Bαj [(` + 1)∗ ]) ∪ Bd Bα|`+1| [(` + 1)∗ ] . We have to show that, in  / Bαn [n∗ ]∪Bαn [(−n)∗ ] for every n > j (the fact that γn Rω (θ) = this case, Rω (θ) ∈ γj Rω (θ) follows as above). When Rω (θ) ∈ Bd(Bαj [(` + 1)∗ ]) this follows from  Definition 3.4(R.3) as before. Assume now that Rω (θ) ∈ Bd Bα|`+1| [(` + 1)∗ ] . Then, by (g), Rω (θ) ∈ / Bn [n∗ ] ∪ B−n[(−n)∗ ] for every n > j. (f ) If ` ≥ 0 then the first two statements of (f) follow directly from the definitions. Moreover, by Remarks 3.2(2) and 3.5(R.1),  diam R (`∗ ) = diam (R(`∗ )) = diam (R((−`)∗ )) = 2 · 2−n` ≤ 2 · 2−(`+1) = 2−` . A SKEW-PRODUCT WITHOUT FIXED-CURVES 21 Assume that ` < 0. From Definition 3.4(R.2) and Remark 3.5(R.2) we get B` [`∗ ] \Bα|`| (`∗ ) ∩ Z|`| ∗ = ∅ and, hence, γ|`| is continuous in an open neighbour-  hood of B` [` ] \Bα|`| (` ) by (a). On the other hand, by (d), θ, γ|`| (θ) ∈ R(`∗ ) for ∗ ∗  every θ ∈ Bd Bα|`| [`∗ ] ⊂ B` [`∗ ] \Bα|`| (`∗ ) . Thus,   R (`∗ ) = R(`∗ ) ∪ Graph γ |`|∗ ∗ B` [` ]\Bα|`| (` ) is closed, connected and projects onto the whole B` [`∗ ] . On the other hand, by (e) and (a) (since ` < 0, |` + 1| = |`| − 1),   γ|`| B` [`∗ ] \Bα|`| (`∗ ) = γ|`|−1 Bα|`+1| [`∗ ] \Bα|`| (`∗ )   ⊂ γ|`|−1 (`∗ ) − 2−n|`|−1 , γ|`|−1 (`∗ ) + 2−n|`|−1 . Thus, by Remark 3.2(1), (c) and Definition 3.4(R.1),   R (`∗ ) = R(`∗ ) ∪ Graph γ|`| B [`∗ ]\B (`∗ ) ` α|`| ∗  ∗ −n|`|  ⊂ Bα|`| [` ] × γ|`|−1 (` ) − 2 , γ|`|−1 (`∗ ) + 2−n|`| ∪    Bα|`+1| [`∗ ] \Bα|`| (`∗ ) × γ|`|−1 (`∗ ) − 2−n|`|−1 , γ|`|−1 (`∗ ) + 2−n|`|−1   ⊂ Bα|`+1| [`∗ ] × γ|`|−1 (`∗ ) − 2−n|`|−1 , γ|`|−1 (`∗ ) + 2−n|`|−1 . Hence, by Definition 3.4(R.1) and Remark 3.5(R.1),  diam R (`∗ ) ≤ 2 · max{α|`+1| , 2−n|`|−1 } = 2 · 2−n|`|−1 ≤ 2 · 2−|`| .  The next results allow us to define the limit pseudo-curve generated by the sequence {(γi , S1 \Zi∗ )}∞ i=0 . Lemma 3.7. The sequence {(γi , S1 \Zi∗ )}∞ i=0 ⊂ PCG is convergent in PCG. Proof. By Proposition 2.9 it suffices to show that {(γi , S1 \Zi∗ )}∞ i=0 is a Cauchy sequence in PCG. By the definition of γi (Definition 3.4) we have  d∞ γi−1 , γi = sup γi−1 (θ) − γi (θ) θ∈S1 \Zi∗ = sup γ (θ) − γ (θ) . i−1 i θ∈(Bαi [i∗ ]\{i∗ })∪(Bαi [(−i)∗ ]\{(−i)∗ }) By Lemmas 3.6(c,d), and Definition 3.4(R.2) and Remark 3.5(R.2), (θ, γi−1 (θ)), (θ, γi (θ)) ∈ R(`∗ ) for θ ∈ Bαi [`∗ ] \{`∗ } and ` ∈ {i, −i}. Hence, by Lemma 3.6(f), d∞ (γi−1 , γi ) ≤ diam(R(i∗ )) = diam(R((−i)∗ )) ≤ 2−i . Since ni is a strictly increasing sequence, for every m ≥ 0, i+m X ∞ X d∞ (γi+m , γi ) ≤ 2−k < 2−(i+1) 1 2k = 2 · 2−(i+1) , k=i+1 k=0 and consequently {(γi , S1 \Zi∗ )}∞ i=0 is a Cauchy sequence in PCG.  Lemma 3.7 allows us to define the following limit pseudo-curve generator of the sequence {(γi , S1 \Zi∗ )}∞ i=0 . 22 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES Definition 3.8. There exists (γ, S1 \O∗ (ω)) ∈ PCG such that (γ, S1 \O∗ (ω)) = lim (γi , S1 \Zi∗ ) i→∞ (that is, γ(θ) = limi→∞ γi (θ) for every θ ∈ S1 \O∗ (ω)). Observe that ∞ \  S1 \O∗ (ω) = S1 \Zi∗ i=1 is a residual set in S . 1 Now, we are ready to define the sequence of pseudo-curves associated to the sequence {(γi , S1 \Zi∗ )}∞ i=0 , and to the limit pseudo-curve generator (γ, S \O (ω)). 1 ∗ This will finally define the pseudo-curve A that we want to construct. Definition 3.9. We denote by Aj := A(γ ,S1 \Z ∗ ) = Graph(γj , S1 \Zj∗ ) j j the pseudo-curve defined by (γj , S1 \Zj∗ ) ∈ PCG, and A = A(γ,S1 \O∗ (ω)) := Graph(γ, S1 \O∗ (ω)). By Definition 3.8 and Proposition 2.11, A = limj→∞ A(γ ,S1 \Z ∗ ) . j j The next lemmas study the properties the pseudo-curves Aj and A. Lemma 3.10. The following statements hold for every ` ∈ Z: (a) Aθn ⊂ R(`∗ )θ for every n ≥ |`| − 1 and θ ∈ Bα|`| [`∗ ] . ∗ ∗ ∗ ∗ ∗ (b) A`n = A`|`| ⊂ R(`∗ )` for every n ≥ |`| . Moreover, A`|`| = R(`∗ )` is a non- degenerate interval. (c) Aθ` = {(θ, γ` (θ)} for every θ ∈ S1 \Z`∗ . (d) A|`| ⊂ S1 × [−1, 1].   Proof. (a) By Lemma 3.6(c,d), Graph γn Bα [`∗ ]\Zn ∗ ⊂ R(`∗ ). Then, the state- |`| ment follows from the compacity of R(`∗ ). (b) From the definition of γi (Definition 3.4) and Definition 3.4(R.2), for every n > |`| there exists an ε(n) > 0 such that γn (θ) = γ|`| (θ) for every θ ∈ Bε(n)(`∗ ) \{`∗ }. ∗ ∗ Hence A`n = A`|`| . Moreover, γ|`| coincides with ϕ`∗ in a neighbourhood of `∗ . Thus, ∗ ∗ A`|`| = R(`∗ )` and it is an interval by Definition 3.1 and Remark 3.2(4). Finally statement (c) follows from Lemma 2.2(a) and Definition 3.9, and (d) from Lemma 3.6(b).  Lemma 3.11. The following statements hold. (a) Aθ ⊂ R(`∗ )θ for every ` ∈ Z and θ ∈ Bα|`| [`∗ ] . ∗ ∗ ∗ (b) A` = A`|`| for every ` ∈ Z. In particular A` is a non-degenerate interval. / O∗ (ω), then Aθ = {(θ, γ(θ))}. (c) If θ ∈ (d) A ⊂ S1 × [−1, 1]. Proof. Statement (c) follows directly from Lemma 2.2(a). Now we prove (a). From Lemma 3.10(a), Aθn ⊂ R(`∗ ) for every ` ∈ Z and n ≥ |`| . On the other hand, by Definition 3.8 and Proposition 2.11, Aθ = limn→∞ Aθn . Hence the result follows from the compacity of R(`∗ ). By Lemma 3.10(b) and the part of the lemma already proved we have ∗ ∗ ∗ A` = lim A`n = A`|`| . n→∞ A SKEW-PRODUCT WITHOUT FIXED-CURVES 23 Statement (d) follows from Lemma 3.10(d), the compacity of S1 × [−1, 1] and the fact that A = limj→∞ Aj .  The next proposition, summarizes the main properties of the set A. Proposition 3.12. The set A is a connected, does not contain any arc of curve and Ω\A has two connected components. Proof. From statements (b) and (c) of the previous lemma, we know that Aθ is connected for every θ ∈ S1 . If A is not connected there exist closed (in A) sets U and V such that U ∩ V = ∅ and U ∪V = A. Observe that π(U )∪π(V ) = π(A) = S1 because every pseudo-curve is a circular set. Moreover, since A is compact, U and V are also compact sets of Ω. Hence, π(U ) are π(V ) compact in S1 . Since S1 is connected, π(U ) ∩ π(V ) 6= ∅. For every θ ∈ π(U ) ∩ π(V ) we have, Aθ = (U ∪ V )θ = U θ ∪ V θ . The sets U θ and V θ are closed, non-empty and disjoint. Consequently, Aθ is not connected; a contradiction. This proves that A is connected. ∗ By Lemma 3.11(b), A` is a non-degenerate interval for every ` ∈ O∗ (ω). Then, since O∗ (ω) is dense in S1 , A does not contain any arc of curve by Lemma 2.3(b). To prove that Ω\A has two connected components we define Ω− := {(θ, y) ∈ Ω : y < min{x ∈ I : (θ, x) ∈ A}}, and Ω+ := {(θ, y) ∈ Ω : y > max{x ∈ I : (θ, x) ∈ A}}. By Lemma 3.11(d) we know that −1 ≤ min{x ∈ I : (θ, x) ∈ A} ≤ max{x ∈ I : (θ, x) ∈ A} ≤ 1. Hence, Ω\A = Ω− ∪ Ω+ , Ω+ and Ω− are disjoint open circular subsets of Ω and Ω− ⊃ S1 × [−2, −1] and Ω+ ⊃ S1 × [1, 2] (in particular, for every θ ∈ S1 , Ωθ+ and Ωθ− are non-degenerate intervals). Thus, Ω+ and Ω− are arc-wise connected and, hence, connected.  4. A collection of auxiliary functions Gi defined on the boxes R (i∗ ) In this section we define a family of auxiliary functions Gi : R(i∗ ) −→ Ω with i ∈ Z and study their properties. In what follows we consider the supremum metric d∞ on the class of all functions F : A −→ Ω with A ⊂ Ω. That is, given F, G : A −→ Ω we set d∞ (F, G) := sup dΩ (F (θ, x), G(θ, x)). (θ,x)∈A In the special case when F and G are skew products with the same base, that is when F (θ, x) = (R(θ), f (θ, x)) and G(θ, x) = (R(θ), g(θ, x)), then d∞ (F, G) := sup |f (θ, x) − g(θ, x)| . (θ,x)∈A Observe that (S(Ω), d∞ ) is a complete metric space. Before defining the maps Gi we need to introduce the necessary notation, and recall and collect some basic facts that we will use in this definition and to study their properties. For every i ∈ Z, we define Mi : Bi [i∗ ] −→ I by Mi (θ) := max{x ∈ I : (θ, x) ∈ R (i∗ )}, and mi : Bi [i∗ ] −→ I by mi (θ) := min{x ∈ I : (θ, x) ∈ R (i∗ )}. The next simple lemma states the basic properties of the maps mi and Mi . 24 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES Lemma 4.1. The following statements hold for every i ∈ Z (a) −1 ≤ mi (θ) ≤ Mi (θ) ≤ 1 for every θ ∈ Bi [i∗ ]. (b) mi and Mi are continuous. (c) mi Bα [i∗ ] and Mi Bα [i∗ ] are piecewise linear. |i| |i| (d) mi (θ) = Mi (θ) = γ|i| (θ) if and only if θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) . Proof. It follows easily from Definition 3.1, the definition of a winged region and Lemma 3.6(b,f).  Notice that, for every i ∈ Z, [ [ R (i∗ ) = R (i∗ )θ = {θ} × [mi (θ), Mi (θ)]. ∗ ∗ θ∈Bi [i ] θ∈Bi [i ] In what follows the interval [mi (θ), Mi (θ)] ⊂ I, defined for every θ ∈ Bi [i∗ ] , will be denoted by Ii,θ . Clearly, for every θ ∈ Bi [i∗ ] , R (i∗ )θ = {θ} × Ii,θ . By Definition 3.4(R.2) and Remark 3.5(R.2), Bi [i∗ ] \{i∗ } ∗ is disjoint from Z|i| . Hence, Lemmas 3.6(a,d) and 3.10(c) can be summarized as:  γ|`| B` [`∗ ]\{`∗ } is continuous,  (2) γ|`| (θ) ∈ I`,θ for every θ ∈ B` [`∗ ] \{`∗ }, and   θ A|`| = {(θ, γ|`| (θ)} for every θ ∈ B` [`∗ ] \{`∗ } for ` ∈ {i, i + 1}. Now we define a family of continuous maps Gi : R (i∗ ) −→ Ω with i ∈ Z, by  Gi (θ, x) = Rω (θ), gi (θ, x) Also, for every θ ∈ Bi [i∗ ] , we will denote the map gi (θ, ·) : Ii,θ −→ I by gi,θ . To define the functions gi,θ , for clarity, we will consider separately two different situations: • i ≥ 0, when R (i∗ ) = R(i∗ ), Bi [i∗ ] = Bα|i| [i∗ ] and Gi (R(i∗ )) strictly contains the smaller box R((i + 1)∗ ), and • i ≤ −1, when Gi (R (i∗ )) is strictly contained in the bigger box R((i+1)∗ ). We start by defining gi,θ for i ≥ 0 in three different ways, depending on the base point θ ∈ Bαi [i∗ ]. In this definition, for simplicity we will use R(i∗ ) instead of R (i∗ ) and Bα|i| [i∗ ] instead of Bi [i∗ ] . Notice that, by Definition 3.4(R.1) and Lemma 3.6(c), for every i ≥ 0 (3) Bδi+1 [i∗ ] ⊂ Bαi+1 (i∗ ) and Bαi+1 [i∗ ] ⊂ Bδi (i∗ ) ⊂ Bαi (i∗ ) , and γi−1 (i∗ ) = ai and γi ((i + 1)∗ ) = ai+1 . Definition 4.2 (Definition of gi for i ≥ 0). ni  θ ∈ Bδi+1 [i∗ ]: gi,θ (x) := γi ((i + 1)∗ ) + 2n2i+1 γi−1 (i∗ ) − x . θ ∈ Bαi+1 [i∗ ] \Bδi+1 (i∗ ): we define gi,θ to be the unique piecewise affine map with two affine pieces, defined on Ii,θ , whose graph  joins mi (θ), Mi+1 Rω (θ) with γi (θ), γi+1  Rω (θ) , and this with the point Mi (θ), m  i+1 Rω (θ) (in particu- lar, gi,θ γi (θ) = γi+1 Rω (θ) ),  θ ∈ Bαi [i∗ ] \Bαi+1 (i∗ ): gi,θ (x) := γi+1 Rω (θ) (that is, gi,θ is constant). A SKEW-PRODUCT WITHOUT FIXED-CURVES 25 The next lemma states the basic properties of the functions Gi for i ≥ 0. Lemma 4.3. The following statements hold for every i ≥ 0 : (a) The map gi,θ is well defined and non-increasing for every θ ∈ Bαi [i∗ ] . Moreover, −1 ≤ gi,θ (x) ≤ 1 for every θ ∈ Bαi [i∗ ] and x ∈ Ii,θ . Furthermore, the function Gi is continuous.  (b) Gi R(i∗ )θ is affine and Gi R(i∗ )θ = R((i + 1)∗ )Rω (θ) for every θ ∈ Bδi+1 [i∗ ] ;  Gi ∗ θ is piecewise affine with two pieces and Gi R(i∗ )θ = R((i + 1)∗ )Rω (θ) R(i ) for every θ ∈ Bαi+1 [i∗ ] \Bδi+1 (i∗ ) ; and  Rω (θ) Gi R(i∗ )θ = Ai+1 for every θ ∈ Bαi [i∗ ] \Bαi+1 (i∗ ) . R (θ) (c) Gi (Aθi ) = Ai+1 ω for every θ ∈ Bαi [i∗ ] . Proof. We will prove all statements of the lemma simultaneously and according to the regions in the definition of the map gi .  Bδ [i∗ ] • We start with the region R(i∗ ) i+1 . Let z ∈ [−δi , δi ] ⊂ R and let θ = i∗ + z ∈ Bδi [i∗ ] . From Definition 3.1 and (3) we get mi (θ) = ai − 2−ni (1 − z) = γi−1 (i∗ ) − 2−ni (1 − z), and (4) Mi (θ) = ai + 2−ni (1 − z) = γi−1 (i∗ ) + 2−ni (1 − z). In a similar way, for every θ ∈ Bδi+1 [i∗ ] (that is, z ∈ [−δi+1 , δi+1 ]), we have Rω (θ) = (i + 1)∗ + z ∈ Bδi+1 [(i + 1)∗ ] , and mi+1 (Rω (θ)) = ai+1 − 2−ni+1 (1 − z) = γi ((i + 1)∗ ) − 2−ni+1 (1 − z), and (5) Mi+1 (Rω (θ)) = ai+1 + 2−ni+1 (1 − z) = γi ((i + 1)∗ ) + 2−ni+1 (1 − z). Hence, for every θ ∈ Bδi+1 [i∗ ] , 2ni gi,θ (mi (θ)) = γi ((i + 1)∗ ) + 2ni+1 2−ni (1 − z) = γi ((i + 1)∗ ) + 2−ni+1 (1 − z) = Mi+1 (Rω (θ)), (6) 2ni gi,θ (Mi (θ)) = γi ((i + 1)∗ ) − 2ni+1 2−ni (1 − z) = γi ((i + 1)∗ ) − 2−ni+1 (1 − z) = mi+1 (Rω (θ)).  So, gi,θ I is the affine map whose graph joins the point mi (θ), Mi+1 (Rω (θ)) i,θ  with Mi (θ), mi+1 (Rω (θ)) . In particular, gi,θ sends the interval Ii,θ affinely onto Ii+1,Rω (θ) or, equivalently, Gi sends the interval R(i∗ )θ affinely onto R((i+1)∗ )Rω (θ) . Then, by Lemma 3.6(b), this implies that −1 ≤ gi,θ (x) ≤ 1 for every x ∈ Ii,θ . Moreover, the continuity of the maps mi , Mi , mi+1 ◦ Rω and Mi+1 ◦ Rω imply that  ∗ gi is well defined and continuous on R(i∗ ) Bδi+1 [i ]  Rω (θ) Next we will prove that Gi Aθi = Ai+1 for every θ ∈ Bδi+1 [i∗ ] . We take θ = i + z ∈ Bδi+1 [i ] \{i }. Then, clearly, z ∈ [−δi+1 , δi+1 ]\{0} ⊂ R. By Definitions 3.4 ∗ ∗ ∗ and 3.1 and statement (3), γi (θ) = ϕi∗ (θ) = ai + 2−ni d = γi−1 (i∗ ) + 2−ni d ∈ Ii,θ , and γi+1 (Rω (θ)) = ϕ(i+1)∗ (θ) = ai+1 − 2−ni+1 d = γi−1 (i∗ ) − 2−ni+1 d ∈ Ii+1,Rω (θ) , where d = (−1)i φ(z). So, for every θ ∈ Bδi+1 [i∗ ] \{i∗ }, 2ni −ni (7) gi,θ (γi (θ)) = γi ((i + 1)∗ ) − 2 d = γi+1 (Rω (θ)). 2ni+1 26 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES Thus, from (3) and (2) we get   Gi Aθi = Gi {(θ, γi (θ))} = {(Rω (θ), gi,θ (γi (θ)))} ω R (θ) = {(Rω (θ), γi+1 (Rω (θ)))} = Ai+1 for every θ ∈ Bδi+1 [i∗ ] \{i∗ }. On the other hand, by the part already proven, gi,i∗ sends the interval Ii,i∗ affinely to Ii+1,(i+1)∗ or, equivalently, Gi sends the interval ∗ ∗ R(i∗ )i = {i∗ } × Ii,i∗ affinely onto R((i + 1)∗ )(i+1) = {(i + 1)∗ } × Ii,(i+1)∗ . This ∗  (i+1)∗  Rω (θ) implies that Gi Aii = Ai+1 by Lemma 3.10(b). Hence, Gi Aθi = Ai+1 for ∗ every θ ∈ Bδi+1 [i ] .  ∗ ∗ • Now we study R(i∗ ) (Bαi+1 [i ]\Bδi+1 (i )) . Observe that Rω (Bα[i∗ ] \{i∗ }) = Bα[(i + 1)∗ ] \{(i + 1)∗ } for α ∈ {αi , αi+1 }. Then, by (2) γi+1 ◦ Rω B [i∗ ]\{i∗ } is continuous, and αi (8) γi+1 (Rω (θ)) ∈ Ii+1,Rω (θ) for every θ ∈ Bαi+1 [i∗ ] \{i∗ }. So, the continuity of the maps mi , Mi , mi+1 ◦ Rω and Mi+1 ◦ Rω imply that gi is  ∗ ∗ well defined and continuous on R(i∗ ) (Bαi+1 [i ]\Bδi+1 (i )) , and  γi (θ), γi+1 (Rω (θ)) ∈ Ii,θ × Ii+1,Rω (θ) for every θ ∈ Bαi+1 [i∗ ] \Bδi+1 (i∗ ) . Consequently, gi,θ maps Ii,θ piecewise affinely with two pieces onto Ii+1,Rω (θ) or, equivalently, Gi sends the interval R(i∗ )θ piece- wise affinely with two pieces onto R((i + 1)∗ )Rω (θ) . Again, by Lemma 3.6(b), this implies that −1 ≤ gi,θ (x) ≤ 1 for every x ∈ Ii,θ . On the other hand, from (3) and (2) we have   Gi Aθi = Gi {(θ, γi (θ))} = {(Rω (θ), gi,θ (γi (θ)))} ω R (θ) = {(Rω (θ), γi+1 (Rω (θ)))} = Ai+1 for every θ ∈ Bαi+1 [i∗ ] \Bδi+1 (i∗ ) .  ∗ ∗ • Finally, we study the region R(i∗ ) (Bαi [i ]\Bαi+1 (i )) . In this case, by definition and Lemma 3.6(b) we have −1 ≤ gi,θ (x) ≤ 1 for every x ∈ Ii,θ . By (8), gi (·, x) = γi+1 ◦ Rω is well defined and continuous in both variables  ∗ ∗ on R(i∗ ) (Bαi [i ]\Bαi+1 (i )) because m and M are continuous. Moreover, for every i i θ ∈ Bαi [i∗ ] \Bαi+1 (i∗ ) and x such that (θ, x) ∈ R(i∗ )θ , we have ω R (θ) {Gi (θ, x)} = {(Rω (θ), gi (θ, x))} = {(Rω (θ), γi+1 (Rω (θ))} = Ai+1 by Definition 3.9 and Lemma 2.2(a). Thus, by Lemma 3.10(a),   Rω (θ) Gi Aθi = Gi R(i∗ )θ = Ai+1 . From all the previous arguments (b) and (c) follow. To end the proof of (a) we have to see that Gi is well defined and globally continuous. This amounts to show that it is well defined on the fibres ∗ R(i∗ )(i ±δi+1 ) = {i∗ ± δi+1 } × Ii,i∗ ±δi+1 and ∗ R(i∗ )(i ±αi+1 ) = {i∗ ± αi+1 } × Ii,i∗ ±αi+1 . We will only show that the two definitions of gi coincide on {θ} × Ii,θ with θ ∈ {i∗ + δi+1 , i∗ + αi+1 }. The case θ ∈ {i∗ − δi+1 , i∗ − αi+1 } follows analogously. A SKEW-PRODUCT WITHOUT FIXED-CURVES 27 We start with θ = i∗ + αi+1 ∈ Bδi (i∗ ) . In this case, Rω (θ) = (i + 1)∗ + αi+1 ∈ Bd(Bαi+1 [(i + 1)∗ ]) and, by Definition 3.1 and Lemma 3.6(c), Mi+1 (Rω (θ)) = mi+1 (Rω (θ)) = a+ i+1 = γi+1 (Rω (θ)).  Thus, the piecewise affine map whose graph joins  the points mi (θ), Mi+1 (Rω (θ)) , γi (θ), γi+1 (Rω (θ)) , and Mi (θ), mi+1 (Rω (θ)) is the constant map γi+1 (Rω (θ)). Hence, gi,θ is well defined for θ = i∗ + αi+1 . ∗ ∗ Now we deal with the case θ =  i + δi+1 ∈ Bδi [i ] . By  (6) and (7) we know that the points mi (θ), Mi+1 (Rω (θ)) , γi (θ), γi+1 (Rω (θ)) and Mi (θ), mi+1 (Rω (θ)) ni  belong to Graph x 7→ γi ((i + 1)∗ ) + 2n2i+1 γi−1 (i∗ ) − x . Consequently, the map ni  γi ((i + 1)∗ ) + 2n2i+1 γi−1 (i∗ ) − x coincides with thepiecewise affine map whose graph joins mi (θ), Mi+1 (Rω (θ)) , γi (θ), γi+1 (Rω (θ)) and Mi (θ), mi+1 (Rω (θ)) . This ends the proof of (a).  Now we define gi,θ for i < 0. In this case, since we are going from a smaller box R (i∗ ) to a bigger one, we only need to define gi,θ in two different ways, depending on the base point θ ∈ Bi [i∗ ]. As in the previous case we need to fix some facts about the elements that we will use in the definition. By Definition 3.4(R.1) and Lemma 3.6(c), for every i < 0 Bδ|i| [(i + 1)∗ ] ⊂ Bα|i| [(i + 1)∗ ] ⊂ Bδ|i+1| ((i + 1)∗ ) ⊂ Bα|i+1| ((i + 1)∗ ) , (9)  Rω Bi [i∗ ] = Bα|i+1| [(i + 1)∗ ] , Bδ|i| [i∗ ] ⊂ Bα|i| (i∗ ) , and γ|i+1| (i∗ ) = ai and γ|i+2| ((i + 1)∗ ) = ai+1 . Consequently, from (2) and Definitions 3.1 and 3.4 we get mi (θ) < γ|i| (θ) < Mi (θ) and mi+1 (Rω (θ)) < γ|i+1| (Rω (θ)) < Mi+1 (Rω (θ)) for every θ ∈ Bα|i| (i∗ ) \{i∗ } (and Rω (θ) ∈ Bα|i| ((i + 1)∗ ) \{(i + 1)∗ }). Then,    m (R (θ)) − γ (R (θ)) M (R (θ)) − γ (R (θ))  i+1 ω |i+1| ω i+1 ω |i+1| ω ei (θ) = min 1, κ n|i| , n|i| >0  2 n (γ (θ) − Mi (θ)) 2 n (γ (θ) − mi (θ))  2 |i+1| |i| 2 |i+1| |i| ei : Bα|i| (i∗ ) \Bδ|i| (i∗ ) −→ (0, 1]. To define the map gi defines a continuous function κ we need an auxiliary function κi : Bα|i| [i∗ ] \Bδ|i| (i∗ ) −→ [0, 1] such that κi is non-decreasing and continuous, κi (i∗ ± δ|i| ) = κ ei (i∗ ± δ|i| ), and ∗ ∗ κi (θ) ≤ κ ei (θ) for every θ ∈ Bα|i| (i ) \Bδ|i| (i ) . In principle any such function would do, but for definiteness, and to show that such function exists, we note that we can take, for instance, ( ei (t) if θ ≤ i∗ − δ|i| , inf t∈[θ,i∗ −δ|i| ]∩Bα|i| (i∗ ) κ κi (θ) = ei (t) if θ ≥ i∗ + δ|i| . inf t∈[i∗ +δ|i| ,θ]∩Bα|i| (i∗ ) κ It is easy to check that this map verifies the desired properties. Definition 4.4 (Definition of gi for i < 0). For every (θ, x) ∈ R (i∗ ) we set  2n|i|    2n|i+1| γ|i+1| (i∗ ) − x + γ|i+2| ((i + 1)∗ ) if θ ∈ Bδ|i| [i∗ ] , n 2 |i|  gi,θ (x) := n κi (θ) γ|i| (θ) − x + γ|i+1| (Rω (θ)) if θ ∈ Bα|i| [i∗ ] \Bδ|i| (i∗ )  2 |i+1| γ|i+1| (Rω (θ)) if θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) . 28 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES The next lemma states the basic properties of the functions Gi for i < 0. Lemma 4.5. The following statements hold for every i < 0 : (a) The map gi,θ is well defined and non-increasing for every θ ∈ Bαi [i∗ ] . Moreover, −1 ≤ gi,θ (x) ≤ 1 for every θ ∈ Bαi [i∗ ] and x ∈ Ii,θ . Furthermore, the function Gi is continuous.  (b) Gi R (i∗ )θ is affine, Gi R (i∗ )θ ⊂ R((i + 1)∗ )Rω (θ) for every θ ∈ Bi [i∗ ] and  Gi R (i∗ )θ = R((i + 1)∗ )Rω (θ) for every θ ∈ Bδ|i| [i∗ ] . R (θ) (c) Gi (Aθ|i| ) = A|i+1| ω for every θ ∈ Bi [i∗ ] . Proof. First we will prove that the map Gi is continuous and that Gi R (i∗ )θ is affine, according to the three regions in the definition.  Bδ [i∗ ]  Bδ [i∗ ] • As in the previous lemma we start with R (i∗ ) |i| = R(i∗ ) |i| . As in the same case of Lemma 4.3, by using (9) instead of (3), it follows that gi,θ I is the affine map whose graph joins the points mi (θ), Mi+1 (Rω (θ)) and i,θ  B [i∗ ] Mi (θ), mi+1 (Rω (θ)) , gi is well defined and continuous on R(i∗ ) δ|i| , gi,θ (γ|i| (θ)) = γ|i+1| (Rω (θ)) for every θ ∈ Bδ|i| [i∗ ] \{i∗ }, Gi sends the interval R(i∗ )θ affinely onto R((i + 1)∗ )θ , and  Rω (θ) Gi Aθ|i| = A|i+1| for every θ ∈ Bδ|i| [i∗ ].   ∗  Bα|i| [i ]\Bδ|i| (i ) ∗  Bα|i| [i ]\Bδ|i| (i ) ∗ ∗ ∗ ∗ • R (i ) = R(i ) . From (2) we know that the maps γ|i| and γ|i+1| ◦ Rω are continuous on the domain Bα|i| [i∗ ] \Bδ|i| (i∗ ) . Hence, the continuity of gi follows from the continuity of the maps κi , mi , Mi , mi+1 ◦ Rω and Mi+1 ◦ Rω . Notice that, from the definition of gi in this region we clearly have that gi,θ (γ|i| (θ)) = γ|i+1| (Rω (θ)), and Gi R = gi (θ, ·) is affine. (i∗ )θ  ∗  Bi [i ]\Bα|i| (i ) ∗ ∗ • R (i ) . In this case we have mi (θ) = γ|i| (θ) = Mi (θ) by definition. Then, the map Gi R (i∗ )θ = gi (θ, ·) is affine because it is constant, and gi is continuous because γ|i| and γ|i+1| ◦ Rω are continuous on the domain Bi [i∗ ] \{i∗ } by (2). To end the proof of (a) we have to see that Gi is well defined and globally continuous. This amounts to show that it is well defined on the fibres ∗ ∗ R(i∗ )(i ±δ|i| ) and R(i∗ )(i ±α|i| ) We start by showing that the two definitions of gi coincide on the fibres R(i∗ )θ for θ ∈ {i∗ ± α|i| }. In this case we have mi (θ) = γ|i| (θ) = Mi (θ). Consequently, Ii,θ = {γ|i| (θ)} and 2n|i|  n|i+1| κi (θ) γ|i| (θ) − x + γ|i+1| (Rω (θ)) = γ|i+1| (Rω (θ)) 2 for x ∈ Ii,θ . Next we consider R(i∗ )θ = {θ} × Ii,θ with θ = i∗ + δ|i| . We will show that the two definitions of gi coincide on this set. The case θ = i∗ − δ|i| follows analogously. A SKEW-PRODUCT WITHOUT FIXED-CURVES 29 For simplicity we will denote δ|i| 2n|i| ∗  ∗ gi,θ (x) := n|i+1| γ|i+1| (i ) − x + γ|i+2| ((i + 1) ), and 2 2n|i|  ξi,θ (x) := n|i+1| γ|i| (θ) − x + γ|i+1| (Rω (θ)). 2 δ Notice that gi,θ|i| is the map gi,θ as defined in the first region while  κi (θ) ξi,θ − γ|i+1| (Rω (θ)) + γ|i+1| (Rω (θ)) is the map gi,θ as defined in the second region. In asimilar way to the previous  δ|i| δ lemma we have that γ|i| (θ), γ|i+1| (Rω (θ)) ∈ Graph gi,θ . Hence, since gi,θ|i| is n|i| δ affine with slope − n2|i+1| , it follows that gi,θ|i| = ξi,θ . So, to end the proof of the 2 lemma, we only have to see that κi (i∗ +δ|i| ) = κ ei (i∗ + δ|i| ) = 1.  Since  thepoints mi (θ),Mi+1 (Rω (θ)) and Mi (θ), mi+1 (Rω (θ)) also belong to δ Graph gi,θ|i| = Graph ξi,θ , it follows that 2n|i|  mi+1 (Rω (θ)) = ξi,θ (Mi (θ)) = n γ (θ) − Mi (θ) + γ|i+1| (Rω (θ)), and 2 |i+1| |i| 2n|i|  Mi+1 (Rω (θ)) = ξi,θ (mi (θ)) = n|i+1| γ|i| (θ) − mi (θ) + γ|i+1| (Rω (θ)). 2 This shows that κei (i∗ + δ|i| ) = κ ei (θ) = 1 and ends the proof of (a). Now we prove (b) according to the three regions in the definition. From the part of the lemma already proven we already know that Gi R (i∗ )θ is affine, and  Gi R (i∗ )θ = R((i + 1)∗ )Rω (θ) for every θ ∈ Bδ|i| [i∗ ] . So, to end the proof of (b) we have to see that (10) gi,θ (Ii,θ ) ⊂ Ii+1,Rω (θ) for every θ ∈ Bi [i∗ ] \Bδ|i| [i∗ ] (by definition, since i < 0, Bi [i∗ ] = Bα|i+1| [i∗ ] ; therefore, Rω (θ) ∈ Bα|i+1| [(i + 1)∗ ] and Ii+1,Rω (θ) = R((i + 1)∗ )Rω (θ) ). For θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) , by (2), we have gi,θ (Ii,θ ) = {γ|i+1| (Rω (θ))} ⊂ Ii+1,Rω (θ) . Now we consider θ ∈ Bα|i| (i∗ ) \Bδ|i| [i∗ ] . Since Mi+1 (Rω (θ)) − γ|i+1| (Rω (θ)) κi (θ) ≤ κ ei (θ) ≤ n|i| , 2 n |i+1| (γ|i| (θ) − mi (θ)) 2 we have 2n|i| Mi+1 (Rω (θ)) − γ|i+1| (Rω (θ))  gi,θ (mi (θ)) ≤ n|i+1| n|i| γ|i| (θ) − mi (θ) + γ|i+1| (Rω (θ)) 2 2 n (γ|i| (θ) − mi (θ)) 2 |i+1| = Mi+1 (Rω (θ)). An analogous computation shows that gi,θ (Mi (θ)) ≥ mi+1 (Rω (θ)). Hence, (10) holds because gi,θ is affine. This ends the proof of (b). Then, by Lemma 3.6(b), Statement (b) of the lemma implies that −1 ≤ gi,θ (x) ≤ 1 for every x ∈ Ii,θ .  Rω (θ) By the part of the lemma already proved we know that Gi Aθ|i| = A|i+1| for every θ ∈ Bδ|i| [i∗ ]. On the other hand, as in the previous lemma, from (9) and (2) 30 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES we get   Gi Aθ|i| = Gi {(θ, γ|i| (θ))} = {(Rω (θ), gi,θ (γ|i| (θ)))} ωR (θ) = {(Rω (θ), γ|i+1| (Rω (θ)))} = A|i+1| for every θ ∈ Bi [i∗ ] \Bδ|i| [i∗ ] . So, (c) holds.  Up to now we have defined the family of auxiliary functions Gi : R (i∗ ) −→ Ω with i ∈ Z. The next step before being able to define the family {Tm } ⊂ S(Ω) is to fix some stratification in the set of boxes R (i∗ ). 5. A stratification in the set of boxes R (i∗ ) In this section we introduce a notion of depth in the set of arcs Bi [i∗ ] defined earlier. This notion introduce a stratification in the set of boxes R (i∗ ) that we study below. Definition 5.1. For every ` ∈ Z we define the depth of `, which will be denoted by depth(`), as the cardinality of the set (see Lemma 3.6(g)) {i ∈ Z : B` [`∗ ] Bi [i∗ ]} = {i ∈ Z : B` [`∗ ] ∩ Bi [i∗ ] 6= ∅} = {i ∈ Z : R (`∗ ) R (i∗ )} = {i ∈ Z : R (`∗ ) ∩ R (i∗ ) 6= ∅}. Also, for every m ∈ Z+ , we denote Dm := {` ∈ Z : depth(`) = m}, D∗m := {i∗ : i ∈ Dm }, and µm := min{|i| : i ∈ Dm }. The next lemma studies the stratification on Z created by the notion of depth. Lemma 5.2. The following statements hold: (a) Dm+1 ⊂ {` ∈ Z : ∃ i ∈ Dm such that B` [`∗ ] Bi [i∗ ]}. (b) For every `, k ∈ Dm it follows that B` [`∗ ] ∩ Bk [k ∗ ] = ∅. Proof. Observe that if B` [`∗ ] Bi [i∗ ] then depth(`) ≥ depth(i) + 1. Hence, (a) holds. Statement (b) follows from Lemma 3.6(g).  In what follows, for every m ∈ Z+ we set [ Bm := Bi [i∗ ] ⊃ D∗m . i∈Dm Note that, by Lemma 5.2(b), Bm is a disjoint union of closed arcs. Therefore, for every θ ∈ Bm , there exists a unique i ∈ Dm such that θ ∈ Bi [i∗ ] . We will denote such integer i by b (θ, m) ∈ Dm . The next two lemmas study the properties of the winged boxes Bi [i∗ ] and R (i∗ ) according to the depth stratification. Lemma 5.4 is the real motivation to introduce the winged boxes. Lemma 5.3. The following statements hold: (a) The sequence {µm }∞ m=0 is strictly increasing. In particular limm→∞ µm = ∞. (b) For every m ∈ Z+ , Bm is dense in S1 , Bm+1 ⊂ Bm and D∗m ∩ Bm+1 = ∅. (c) O∗ (ω) ⊂ B0 , and Aθ = {(θ, 0)} for every θ ∈ S1 \B0 . (d) Let i ∈ Z and θ ∈ Bi [i∗ ] \Bdepth(i)+1 . Then, θ ∈ / O∗ (ω) unless θ = i∗ , and Aθn = Aθ|i| for every n ≥ |i| . In particular Aθ = Aθ|i| . A SKEW-PRODUCT WITHOUT FIXED-CURVES 31 Proof. By Lemmas 5.2(a) and 3.6(g) it follows that for every m ∈ Z+ and ` ∈ Dm+1 there exists i ∈ Dm such that B` [`∗ ] Bi [i∗ ] and |i| < |`| . Thus, Bm+1 ⊂ Bm and µm < µm+1 . This proves (a) and the second statement of (b). / Bm+1 for every i ∈ Dm . Assume by way of contradic- Next we will show that i∗ ∈ tion that there exists i ∈ Dm such that i∗ ∈ Bm+1 . Let k = b (i∗ , m + 1) ∈ Dm+1 . Clearly, i 6= k and i∗ ∈ Bk [k ∗ ] . Then, by Lemma 3.6(g), |k| < |i| and Bi [i∗ ] Bk [k ∗ ] . Thus, m = depth(i) ≥ depth(k) + 1 = m + 2; a contradiction. Now we prove the first statement of (c). From the definitions and the part of (b) already proven we have [ ∞ [ O∗ (ω) ⊂ Bi [i∗ ] ⊂ Bm = B0 . i∈Z m=0 To end the proof of (b) it remains to show the density of Bm . We will do it by induction on m. Clearly B0 ⊃ O∗ (ω) is dense in S1 because so is O∗ (ω). Suppose that (b) holds for Bm . We will show that (b) also holds for Bm+1 . Choose θ ∈ Bm and set i = b (θ, m) . Since O∗ (ω) is dense in S1 , there exists a sequence {sn }∞ n=0 ⊂ Z such that s∗n ∈ Bi (i∗ ) and limn→∞ s∗n = θ. As above, we get that depth(sn ) ≥ depth(i) + 1 = m + 1. Moreover, s∗n ∈ Bdepth(sn ) ⊂ Bm+1 for every n. Consequently, Bm ⊂ Bm+1 , and the density of Bm+1 follows from the density of Bm . Next we prove the second statement of (c). From above it follows that [ [ Bα|i| [i∗ ] ⊂ Bi [i∗ ] ⊂ B0 . i∈Z i∈Z Hence, by the definition of the maps γm (Definition 3.4) it follows that γm (θ) = / B0 and m ∈ Z+ . So, γ(θ) = limm→∞ γm (θ) = 0, and γ0 (θ) = 0 for every θ ∈ θ A = {(θ, γ(θ))} = {(θ, 0)} by Lemma 3.11(c). This ends the proof of (c). (d) If θ = i∗ then the statement follows from Lemmas 3.10(b) and 3.11(b). So, we assume that θ 6= i∗ . ∗ By Definition 3.4(R.2) and Remark 3.5(R.2) we get that θ ∈ / Z|i|+1 . Hence, if θ ∈ O (ω), it follows that θ = k ∈ Bdepth(k) with |k| > |i| + 1 and Bk [k ∗ ] ∩ Bi [i∗ ] 6= ∗ ∗ ∅. Thus, by Lemma 3.6(g), depth(k) ≥ depth(i) + 1. By (b), this implies that θ = k ∗ ∈ Bdepth(i)+1 ; a contradiction. Therefore, θ ∈ / O∗ (ω). On the other hand, ∗ θ∈ / B−i[(−i) ] by Definition 3.4(R.2). / Bα|k| [k ∗ ] for every k ∈ Z such that |k| > |i| , then γn (θ) = γ|i| (θ) and If θ ∈ Aθn = Aθ|i| for every n ≥ |i| , by Definition 3.4 and Lemma 3.10(c). Now assume that θ ∈ Bα|k| [k ∗ ] for some k ∈ Z such that |k| > |i| and |k| is minimal with these properties. If θ ∈ Bk (k ∗ ) , as above we get that depth(k) ≥ depth(i) + 1 and θ ∈ Bdepth(k) ⊂ Bdepth(i)+1 . Thus, θ ∈ Bd(Bk [k ∗ ]) = Bd(Bα|k| [k ∗ ]) and k ≥ 0. So, by Lemma 3.6(c) and the definition of the maps γj (Definition 3.4), γ|k| (θ) = γ|k|−1 (θ). Moreover, by Lemma 3.6(e), γj (θ) = γ|k| (θ) for every j > |k| . On the other hand, the minimality of |k| implies that θ ∈ / Bα|`| [`∗ ] for every ` ∈ Z such that |k| > |`| > |i| . Hence, by the definition of the maps γj (Definition 3.4), γj (θ) = γ|i| (θ) for every |k| > j > |i| . In short, we have proved that γj (θ) = γ|i| (θ) for every j ≥ |i| . Thus, as above, Aθn = Aθ|i| for every n ≥ |i| . This ends the proof of the lemma.  32 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES Lemma 5.4. Assume that Bi [i∗ ] ⊂ Bk [k ∗ ] for some i ∈ Dm , k ∈ Dm−1 and m ∈ N. Then, |k| < |i| and |k + 1| < |i + 1| unless k ≥ 0 and i = −(k + 2) (whence |k + 1| = |i + 1|). Moreover, the following statements hold: (a) For every θ ∈ Bi [i∗ ] , γ|k| (θ) = γ|k|+1 (θ) = · · · = γ|i|−1 (θ) ∈ Ii,θ and, when |k + 1| < |i + 1|, γ|k+1| (Rω (θ)) = γ|k+1|+1 (Rω (θ)) = · · · = γ|i+1|−1 (Rω (θ)) (b) For every θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) , γ|i| (θ) = γ|i|−1 (θ) and Ii,θ = {γ|i| (θ)} = {γ|k| (θ)} ⊂ Ik,θ . Proof. The fact that |k| < |i| follows from Lemma 3.6(g). Therefore, either |k + 1| < |i + 1| or k ≥ 0, i = −(k + 2) and |k + 1| = |i + 1| or k ≥ 0, i = −(k + 1) and |k + 1| > |i + 1|. In the last case, Bi [i∗ ] = B−(k+1)[(−(k + 1))∗ ] and Bk [k ∗ ] must be disjoint by Definition 3.4(R.2) (with j = k); which is a contradiction. Thus |k + 1| < |i + 1| unless k ≥ 0 and i = −(k + 2) (|k + 1| = |i + 1|). By Definition 3.4(R.2) and Remark 3.5(R.2), Bi [i∗ ] ∩ Z|i|−1 ∗ = ∅. Hence, from the definition of the maps γj (Definition 3.4), to prove that γ|k| B [i∗ ] = γ|k|+1 B [i∗ ] = · · · = γ|i|−2 B [i∗ ] = γ|i|−1 B [i∗ ] , i i i i it is enough to show that Bα|`| [`∗ ] ∩ Bi [i∗ ] = ∅ for every ` such that |k| < |`| < |i| . Assume that Bα|`| [`∗ ] ∩ Bi [i∗ ] 6= ∅ for some ` such that |k| < |`| < |i| . Then, 6 Bα|`| [`∗ ] ∩ Bi [i∗ ] ⊂ B` [`∗ ] ∩ Bi [i∗ ] ⊂ B` [`∗ ] ∩ Bk [k ∗ ] ∅= and, by Lemma 3.6(g), Bi [i∗ ] B` [`∗ ] Bk [k ∗ ] . So, in a similar way as before, m = depth(i) ≥ depth(`) + 1 ≥ depth(k) + 2 = m + 1; a contradiction. This ends the proof of the first statement of (a). Now we show that if |k + 1| < |i + 1| − 1, then γ|k+1| (Rω (θ)) = γ|k+1|+1 (Rω (θ)) = · · · = γ|i+1|−1 (Rω (θ)) , and are well defined. First we prove that γ` (Rω (θ)) is well defined for every ` = 0, 1, . . . , |i + 1| − 1. For every θ ∈ Bi [i∗ ] we have (  Bαi [(i + 1)∗ ] when i ≥ 0, and Rω (θ) ∈ Rω Bi [i∗ ] = ∗ ∗ Bα|i+1| [(i + 1) ] ⊂ Bi+1[(i + 1) ] when i < 0. In any case, by Definition 3.4(R.2) and Remark 3.5(R.2) with j = i when i ≥ 0 and ` = −(j + 1) = i + 1 when i < 0, and Lemma 3.6(a), ( Zi∗ when i ≥ 0, and Rω (θ) ∈ / ∗ Z|i+1|−1 when i < 0, ∗ ∗ and γ` (Rω (θ)) is well defined for ` = 0, 1, . . . , |i + 1| − 1 (recall that Zm ⊂ Zm+1 for every m ≥ 0). Now, assume by way of contradiction that γ` (Rω (θ)) 6= γ`−1 (Rω (θ)) for some ` ∈ {|k + 1| + 1, |k + 1| + 2, . . . , |i + 1| − 1}, A SKEW-PRODUCT WITHOUT FIXED-CURVES 33 and ` is minimal with this property (observe that ` ≥ 1). By the definition of the map γ` (Definition 3.4), Rω (θ) ∈ Bα` ((q + 1)∗ ) with q ∈ {` − 1, −(` + 1)} and, hence, θ ∈ Bα` (q ∗ ) . Since |k + 1| + 1 ≤ ` < |i + 1|, when q = −(` + 1) ≤ −2, |k + 1| + 2 ≤ −q ≤ |i + 1| and Bα` (q ∗ ) = B−(`+1) ((−(` + 1))∗ ) = Bq (q ∗ ) . Otherwise, when q = ` − 1 ≥ 0, |k + 1| ≤ q ≤ |i + 1| − 2 and Bα` (q ∗ ) ⊂ Bα`−1 ((` − 1)∗ ) = B`−1 ((` − 1)∗ ) = Bq (q ∗ ) , by Definition 3.4(R.1). Next we want to use Lemma 3.6(g) to show that Bi [i∗ ] Bq [q ∗ ] Bk [k ∗ ] . To this end we have to compare |q| with |i| and |k| . Notice Bq [q ∗ ] ∩ Bk [k ∗ ] 6= ∅ because θ ∈ Bq (q ∗ ) ∩ Bi [i∗ ] ⊂ Bq (q ∗ ) ∩ Bk [k ∗ ] . If k ≥ 0, |q| ≥ |k + 1| > |k| . When k, q < 0, |q| ≥ |k + 1|+2 = |k|+1 > |k| . If k < 0 and q ≥ 0, |q| = q ≥ |k + 1| = |k|−1. If q = |k|−1 (that is, k = −(q +1)), as above, by Definition 3.4(R.2) with j = q we get Bk [k ∗ ] ∩ Bq [q ∗ ] = ∅; a contradiction. So, |q| > |k| unless |q| = |k| and k < 0 ≤ q. Summarizing, we have shown that |q| ≥ |k| and q 6= k. Then, from Lemma 3.6(g) we get that |q| > |k| and Bq [q ∗ ] Bk [k ∗ ] . Now we will study the relation of Bq [q ∗ ] with the box Bi [i∗ ] . From above we get that Bq [q ∗ ] ∩ Bi [i∗ ] 6= ∅. If i < 0, |q| ≤ |i + 1| = |i| − 1. When q, i ≥ 0, we have |q| = q ≤ |i + 1| − 2 = |i| − 1. If i ≥ 0 and q < 0, |q| ≤ |i + 1| = |i| + 1. Assume that i ≥ 0 and q = −(i + 1) < 0. In this case, additionally, q = −(` + 1) and, thus, i = ` ≥ 1. Then,  Rω (θ) ∈ Rω Bi [i∗ ] = Rω (Bαi [i∗ ]) = Bαi [(i + 1)∗ ] , and Rω (θ) ∈ Bα` ((q + 1)∗ ) = Bαi ((−i)∗ ) ⊂ B−i ((−i)∗ ) , which is a contradiction by Definition 3.4(R.2). Summarizing, |q| < |i| unless |q| = |i| and q < 0 ≤ i (that is, |q| ≤ |i| and q 6= i). Then, again by Lemma 3.6(g), |q| < |i| and Bi [i∗ ] Bq [q ∗ ] Bk [k ∗ ] . So, as before, m = depth(i) ≥ depth(q) + 1 ≥ depth(k) + 2 = m + 1; a contradiction. This ends the proof of (a). Now we assume that θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) . By Lemmas 3.6(e) and 4.1(d), γ|i| (θ) = γ|i|−1 (θ) and Ii,θ = {γ|i| (θ)} = {γ|i|−1 (θ)} = {γ|k| (θ)}. On the other hand, by Lemma 5.3(b), D∗m−1 ∩ Bm = ∅ which implies that θ 6= k ∗ because k ∗ ∈ D∗m−1 and θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) ⊂ Bm . So, by (2), Ii,θ = {γ|i| (θ)} = {γ|k| (θ)} ⊂ Ik,θ . Now we prove that γ|i|−1 (θ) ∈ Ii,θ for every θ ∈ Bi [i∗ ] . From above, we have Ii,θ = {γ|i|−1 (θ)} for every θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) . Moreover, when θ ∈ Bα|i| (i∗ ) the statement follows directly from Lemma 3.6(c). Thus, (b) is proved.  34 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES 6. Boxes in the wings To prove Theorem A we will inductively construct a Cauchy sequence {Tm }∞ m=0 ⊂ S(Ω) that gives the function T from Theorem A as a limit. This section is devoted to study the points in the wings of boxes in the circle and its interaction with boxes of higher depth. The resulting technology is necessary to be able to construct the sequence {Tm }∞ m=0 so that it is Cauchy sequence. Unfor- tunately this will complicate even more the definition of the functions Tm and the proof of its continuity. We start by introducing some more notation. For every m ∈ Z+ we set [ Bm := Bα|i| [i∗ ] ⊂ Bm , and i∈Dm n o WDBm := θ ∈ Bm \Bm : θ ∈ Bj for some j > m . On the other hand, the smallest number j from the above definition will be called the least essential depth of θ below m, and will be denoted by led (θ, m) . That is, led (θ, m) denotes the positive integer larger than m such that θ ∈ Bj \Bj for j = m, m + 1, . . . , led(θ, m) − 1 and θ ∈ Bled(θ,m) . The following simple lemmas are useful to better understand and use the above definitions. The next lemma establishes the relation between boxes in the wings of increasing depth. Lemma 6.1. Assume that θ ∈ WDBm for some m ∈ Z+ and set ` = led (θ, m) . Then, the following statements hold. (a) For every j = m, m + 1, . . . , ` the numbers ij = b (θ, j) ∈ Dj are well defined and are all of them negative except, perhaps, i` = b (θ, led (θ, m)) . (b) |im | < |im+1 | < · · · < |i`−1 | < |i` | , and θ ∈ Bα|i | [(i` )∗ ] ⊂ Bi`−1 ((i`−1 )∗ ) \Bα|i [(i`−1 )∗ ] ` `−1 | ⊂ Bi`−2 ((i`−2 )∗ ) \Bα|i [(i`−2 )∗ ] ⊂ · · · ⊂ Bim ((im )∗ ) \Bα|im | [(im )∗ ] . `−2 | (c) For every j = m, m + 1, . . . , ` − 1, Bα|i | [(i` )∗ ] ⊂ WDBj , led (ν, j) = led (θ, m) ` and b (ν, led (ν, j)) = b (θ, led (θ, m)) = i` for every ν ∈ Bα|i | [(i` )∗ ] . ` (d) Iim ,ν = {γ|im | (ν)} ⊂ Ii` ,ν for every ν ∈ Bα|i | ((i` )∗ ) and ` Iim ,ν = {γ|im | (ν)} = {mi` (ν)} = {Mi` (ν)} = {γ|i | (ν)} = Ii` ,ν `   for every ν ∈ Bd Bα|i | [(i` )∗ ] . ` Proof. Since Bi [i ] = Bαi [i∗ ] for every i ≥ 0, ∗ [  (11) Bm \Bm = Bi [i∗ ] \Bα|i| [i∗ ] i∈Dm i<0 for every m ∈ Z+ . Statement (a) follows from Lemma 5.3(b) and (11). Then, (b) follows from Lemma 3.6(g). Statement (c) is an easy consequence of (b) and the definitions. Now we prove (d) iteratively. Fix ν ∈ Bα|i | ((i` )∗ ) . By (b) ` ν ∈ Bim+1 ((im+1 )∗ ) \Bα|i [(im+1 )∗ ] ⊂ Bim ((im )∗ ) \Bα|im | [(im )∗ ] m+1 | A SKEW-PRODUCT WITHOUT FIXED-CURVES 35 provided that ` = led (θ, m) > m + 1. Hence, by Lemmas 4.1(d) and 5.4, γ|im | (ν) = γ|im |+1 (ν) = · · · = γ|i (ν), and m+1 | Iim ,ν = {γ|im | (ν)} = {γ|i (ν)} = Iim+1 ,ν . m+1 | By iterating this argument we get, γ|im | (ν) = γ|im |+1 (ν) = · · · = γ|i (ν) and Iim ,ν = Ii`−1 ,ν . `−1 | Again by (b) and Lemmas 4.1(d) and 5.4, γ|im | (ν) = γ|im |+1 (ν) = · · · = γ|i | (ν) and Iim ,ν = Ii` ,ν `   when ν ∈ Bd Bα|i | [(i` )∗ ] and, otherwise, ` γ|im | (ν) = γ|im |+1 (ν) = · · · = γ|i (ν) and Iim ,ν ⊂ Ii` ,ν . ` −1|  Equipped with above results and definition we are going to define two maps, analogous to the maps mi and Mi , on the wings of the negative boxes. Definition 6.2. For every m ∈ Z+ we define WFDm := {b (θ, led(θ, m)) : θ ∈ WDBm } ⊂ Z, [ WIBm := Int(WDBm ) = Bα|i| (i∗ ) , i∈WFDm [  WBm := Bi [i∗ ] \Bα|i| (i∗ ) , and i∈Dm i<0 [  EBm := Bd Bi [i∗ ] ⊂ Bm . i∈Dm By Lemma 6.1(a,c), WFDm is well defined and WIBm ⊂ WDBm ⊂ Bm \Bm ⊂ WBm . Consequently, Bm = Bm ∪ WBm . Then, we can define functions τm : WBm −→ I and λm : WBm −→ I as follows: ( Mb (θ,led(θ,m)) (θ) if θ ∈ WIBm , τm (θ) := γ|b (θ,m)| (θ) otherwise, ( mb (θ,led(θ,m)) (θ) if θ ∈ WIBm , λm (θ) := γ|b (θ,m)| (θ) otherwise. Clearly, by Lemmas 4.1(a) and 3.6(b), −1 ≤ λm (θ) ≤ τm (θ) ≤ 1 for every θ ∈ WBm . So, we can define IWm,θ := [λm (θ), τm (θ)] ⊂ [0, 1]. The next lemmas will help us in the definition and study of the maps Tm . Lemma 6.3. The following statements hold for every m ∈ Z+ . (a) WIBm ∩ Bm = WIBm ∩ EBm = ∅. 36 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES   (b) Let θ ∈ WBm . Then, Ib (θ,m),θ = γ (θ) , |b (θ,m)| Ib (θ,m),θ = IWm,θ / WIBm , and when θ ∈ Ib (θ,m),θ ⊂ IWm,θ when θ ∈ WIBm . (c) Assume that m ∈ N and let U be a connected component of WBm such that U ⊂ WBm−1 . Then, WDBm ∩ U ⊂ WDBm−1 , WIBm ∩ U = WIBm−1 ∩ U and IWm,θ = IWm−1,θ for every θ ∈ U. Proof. (a) By Lemma 6.1(b),    θ ∈ Bb (θ,m) (b (θ, m))∗ \Bα|b (θ,m)| (b (θ, m))∗ and b (θ, m) < 0 for every θ ∈ WIBm ⊂ WDBm . So, by Lemma 5.2(b), we get θ∈/ Bm ∪ EBm . n o (b) The fact that Ib (θ,m),θ = γ|b (θ,m)| (θ) follows from Lemma 4.1(d). The other two statements follow from Definition 6.2 and Lemma 6.1(d). (c) The assumption that U is a connected component of WBm and U ⊂ WBm−1 implies by Lemmas 5.2(b) and 3.6(g) that there exist i ∈ Dm and k ∈ Dm−1 , i, k < 0, such that U is a connected component of Bi [i∗ ] \Bα|i| (i∗ ) ⊂ Bk (k ∗ ) \Bα|k| [k ∗ ] ⊂ WBm−1 . Again by Lemma 5.2(b) this implies that U ⊂ Bm−1 \Bm−1 . Moreover, by definition, WDBm ⊂ Bm \Bm . Consequently, WDBm ∩ U ⊂ WDBm−1 . Let θ ∈ WIBm ∩ U ⊂ WDBm ∩ U ⊂ WDBm−1 ∩ U. By Definition 6.2 and Lemma 6.1(a,b), i = b (θ, m) and there exists ` = b (θ, led (θ, m)) ∈ WFDm such that θ ∈ Bα|`| (`∗ ) ⊂ Bi [i∗ ] \Bα|i| (i∗ ) ⊂ Bk (k ∗ ) \Bα|k| [k ∗ ] . Therefore, again by Lemma 6.1(a–c) and Definition 6.2, led (θ, m − 1) = led (θ, m) , ` = b (θ, led(θ, m)) = b (θ, led(θ, m − 1)) ∈ WFDm−1 and θ ∈ Bα|`| (`∗ ) ⊂ WIBm−1 . Hence, WIBm ∩ U ⊂ WIBm−1 . Now assume that θ ∈ WIBm−1 ∩ U. As above, there exist r = b (θ, m) ∈ Dm and ` = b (θ, led (θ, m − 1)) ∈ WFDm−1 such that θ ∈ Bα|`| (`∗ ) ⊂ Br (r∗ ) \Bα|r| [r∗ ] ⊂ Bk (k ∗ ) \Bα|k| [k ∗ ] . Since θ ∈ U ⊂ Bi [i∗ ], Lemma 5.2(b) gives i = r and θ ∈ Bα|`| (`∗ ) ⊂ U. Moreover, by Lemma 6.1(c), ` = b (θ, led (θ, m − 1)) = b (θ, led (θ, m)) ∈ WFDm and, so, θ ∈ Bα|`| (`∗ ) ⊂ WIBm . Thus, WIBm ∩ U = WIBm−1 ∩ U. To end the proof of the lemma we have to show that IWm,θ = IWm−1,θ for every θ ∈ U. Assume first that θ ∈ U \WIBm ⊂ WBm \WIBm . Then, θ ∈ U \WIBm = U \WIBm−1 ⊂ WBm−1 \WIBm−1 and, by (b) and Lemmas 4.1(d) and 5.4, IWm,θ = Ii,θ = {γ|i| (θ)} = {γ|k| (θ)} = Ik,θ = IWm−1,θ . If θ ∈ U ∩ WIBm = U ∩ WIBm−1 then we get   IWm,θ = mb (θ,led(θ,m)) (θ), Mb (θ,led(θ,m)) (θ)   = mb (θ,led(θ,m−1)) (θ), Mb (θ,led(θ,m−1)) (θ) = IWm−1,θ from Definition 6.2 and Lemma 6.1(c).  A SKEW-PRODUCT WITHOUT FIXED-CURVES 37 Lemma 6.4. Let m ∈ Z and let U be a connected component of WBm . Then, the + functions λm U and τm U are continuous. Proof. We will prove only the continuity of λ m U . The proof of the continuity of τm U is analogous. By Lemmas 6.1(c) and 4.1(b) we get for every ` ∈ WFDm , ` = b (ν, led(ν, m)) for every ν ∈ Bα|`| [`∗ ] , and the (12) function m` is continuous on Bα|`| [`∗ ] . Let ` ∈ WFDm be such that Bα|`| (`∗ ) ⊂ WIBm ∩ U. Thus, by (12), the function λm = m` is continuous on Bα|`| (`∗ ) . So, we have to show that λm is continuous at every θ ∈ U \WIBm . To show this we will use a simple usual ε–δ game. Fix ε > 0. By Lemma 5.2(b) it follows that U is a connected component of Bi [i∗ ] \Bα|i| (i∗ ) for some i ∈ Dm , i < 0, and (13) b (ν, m) = i for every ν ∈ U. By Lemma 3.6(a) and Definition 3.4(R.2) and Remark 3.5(R.2), the function γ|i| U is continuous. So, there exists δ |i| = δ |i| (θ) > 0 such that γ|i| (θ), γ|i| (ν) < ε/2 provided that (14) dS1 (θ, ν) < δ |i| . On the other hand, by (12), e m` (ν) < ε/2 for every ` ∈ WFDm , there exists δ` > 0 such that m` (θ), (15)  for every θe ∈ Bd Bα|`| [`∗ ] and ν ∈ Bd Bα|`| [`∗ ] such that dS1 (θ, ν) < δ` . Now we will define δ. Note that there exists N ∈ N such that 2−N < ε/2. Then we set:  δ = δ(θ) := min δ |i| (θ), min{δ` : ` ∈ WFDm and |`| < N } . Clearly, δ > 0 because the set {` ∈ WFDm : |`| < N } is finite. To end the proof of the lemma we have to show that |λm (θ) − λm (ν)| < ε whenever ν ∈ U and dS1 (θ, ν) < δ. Assume that ν ∈ U and dS1 (θ, ν) < δ (recall that we have the assumption that / WIBm ). If ν ∈ θ∈ / WIBm , then dS1 (θ, ν) < δ ≤ δ |i| (θ) and, by (13) and (14), |λm (θ) − λm (ν)| = γ (θ) − γ (ν) < ε/2 < ε. |i| |i| Now assume that there exists ` ∈ WFDm such that ν ∈ Bα|`| (`∗ ) ⊂ WIBm .  Clearly, there exists θe ∈ Bd Bα|`| [`∗ ] such that e < d 1 (θ, ν) < δ ≤ δ |i| (θ) and dS1 (θ, θ) S e ν) < d 1 (θ, ν) < δ. dS1 (θ, S Observe that, by Lemma 5.2(b), θe ∈ / WIBm . Hence, by (13) and Lemma 6.1(c,d), e = γ (θ) λm (θ) e = m` (θ). e |i| e ν) < δ ≤ δ` and, by (15), m` (θ) If |`| < N, then dS1 (θ, e − m` (ν) < ε/2. Other- wise, by Lemma 3.6(f), e − m` (ν) < diam (R(`∗ )) ≤ 2−|`| ≤ 2−N < ε/2. m` (θ) 38 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES e − m` (ν) < ε/2. Thus, again by (13) and (14), In any case, m` (θ) e − λm (ν) e + λm (θ) |λm (θ) − λm (ν)| ≤ λm (θ) − λm (θ) e − m` (ν) < ε. e + m` (θ) = γ|i| (θ) − γ|i| (θ)  7. A Cauchy sequence of skew products. Proof of Theorem A In this section prove Theorem A. To do this we inductively construct a Cauchy sequence {Tm }∞ m=0 ⊂ S(Ω) that gives the function T from Theorem A as a limit. The sequence {Tm }∞ m=0 ⊂ S(Ω) is defined so that Tm (θ, x) = (Rω (θ), fm (θ, x)) and fm : Ω −→ I is continuous in both variables. To build these functions we will use the auxiliary functions Gi : R(i∗ ) −→ Ω with i ∈ Z from Section 4. The maps fm (θ, ·) will also be denoted as fm,θ , and will be defined non-increasing, and such that fm,θ (2) = −2 and fm,θ (−2) = 2 for every θ ∈ S1 . To make more evident the strategy of the construction of this sequence of maps we will separate several cases, and we will state without proofs the results that study these maps. After establishing all the definitions and results related to the construc- tion of the sequence {Tm }∞ m=0 without having been distracted by the technicalities involving the proofs, we will proceed to provide the missing proofs. More precisely, we will start by defining the map T0 and stating without proof the proposition that summarizes the necessary properties of this map. Next we will inductively define the maps {Tm }∞ m=1 ⊂ S(Ω) and state without proof the proposition that establishes the properties of the whole sequence {Tm }∞ m=0 . Then, as we have said, we prove Theorem A and in the next three sections we will provide all pending proofs. In what follows C(I, I) will denote the class of all continuous maps from I to itself. We endow C(I, I) with the supremum metric denoted by k·k so that (C(I, I), k·k) is a complete metric space. Next we define the map T0 . Definition 7.1 (The map T0 ). Assume first that θ ∈ B0 and let i = b (θ, 0) (that is θ ∈ Bi [i∗ ]). In this case we set:    gi,θ (x) if x ∈ Ii,θ ,   g (m (θ))−2 i,θ i f0,θ (x) = mi (θ)+2 (x + 2) + 2 if x ∈ [−2, mi (θ)],     gi,θ (Mi (θ))+2 (x − 2) − 2 Mi (θ)−2 if x ∈ [Mi (θ), 2]. If θ ∈ S1 \B0 then we define f0,θ to be the unique piecewise affine map with two affine pieces whose graph joins the point (−2, 2) with (0, γ(Rω (θ))), and this with the point (2, −2). Next we introduce some more notation to be able to define the maps {Tm }∞ m=1 . For every k ∈ Z we set Vk∗ :=  Bk [k ∗ ] = Bk [k ∗ ] × I and, for every m ∈ Z+ , [ Vm :=  Bm = Bm × I = Vi∗ . i∈Dm A SKEW-PRODUCT WITHOUT FIXED-CURVES 39 Definition 7.2 (The maps Tm with m > 0). Now we assume that we have defined the function Tm−1 for some m ≥ 1 and we define Tm (θ, x) = (Rω (θ), fm (θ, x)) as follows. By Lemma 5.2(b), for every (θ, x) ∈ Vm , we have θ ∈ Bi [i∗ ] ⊂ Bm with i = b (θ, m) ∈ Dm (and, of course, x ∈ I). Then we define:    fm−1,θ (x) if θ ∈ S1 \Bm ; x ∈ I,     gi,θ (x) if θ ∈ Bm ; x ∈ Ii,θ ,      2−gi,θ (mi (θ)) (f  if θ ∈ Bm ; x ∈ [−2, mi (θ)],   2−fm−1,θ (mi (θ)) m−1,θ (x) − 2) + 2   2+g (Mi (θ)) i,θ fm,θ (x) = 2+fm−1,θ (Mi (θ)) (fm−1,θ (x) + 2) − 2 if θ ∈ Bm ; x ∈ [Mi (θ), 2],     γ|i+1| (Rω (θ)) if θ ∈ WBm ; x ∈ IWm,θ ,     2−γ|i+1| (Rω (θ))   if θ ∈ WBm ; x ∈ [−2, λm (θ)],   2−fm−1,θ (λm (θ)) (fm−1,θ (x) − 2) + 2     2+γ (Rω (θ)) |i+1| 2+fm−1,θ (τm (θ)) (fm−1,θ (x) + 2) − 2 if θ ∈ WBm ; x ∈ [τm (θ), 2]. Since Vm ⊂ Vm−1 , fm−1,θ is defined on Vm . Moreover, the above formula defines fm,θ for every θ ∈ Bm since, by Definition 6.2, Bm = Bm ∪ WBm . We also remark that fm,θ formally is defined in two different ways when θ ∈ WBm ∩ Bm . Later on we will show that fm,θ is well defined. The next proposition studies the maps {Tm }∞ m=0 and describes their properties. Proposition 7.3. The following statements hold for every m ∈ Z+ . (a) The map Tm is well defined, continuous and belongs to S(Ω). (b) For every θ ∈ S1 , fm,θ is non-increasing,   = −2, fm,θ (−2) = 2. and fm,θ (2) Moreover, −1 ≤ f0,θ Mb (θ,m) (θ) ≤ f0,θ mb (θ,m) (θ) ≤ 1 for every θ ∈ Bm .  ∗ (i+1)∗ (c) For every i ∈ Dm , Tm R (i∗ ) = Gi , Tm Ai|i| = A|i+1| , and Tk {i∗ }×I = Tm {i∗ }×I (that is, fk,i∗ = fm,i∗ ) for every k > m. The next result shows that the sequence {Tm }∞ m=0 has a limit in S(Ω). Proposition 7.4. For every m ≥ 2 and θ ∈ S1 , (16) kfm,θ − fm−1,θ k ≤ 2 · 2−|b (θ,m−1)| . Moreover, the sequence {Tm }∞ k=0 is a Cauchy sequence. Finally we are ready to prove the main result of the paper. It follows from the next result which gives a more concrete version of Theorem A. Theorem 7.5. There exists a map T ∈ S(Ω) with f (θ, ·) non-increasing for every θ ∈ S1 , such that T permutes the upper and lower circles of Ω (thus having a periodic orbit of period two of curves), and there exists a connected pseudo-curve A ⊂ Ω which does not contain any arc of a curve such that T (A) = A and there does not exist any T -invariant curve. Proof. By Propositions 7.3 and 7.4, there exists a map T (θ, x) = (Rω (θ), f (θ, x)) = (Rω (θ), lim fm (θ, x)) ∈ S(Ω) m→∞ with f (θ, ·) non-increasing for every θ ∈ S1 such that T permutes the upper and lower circles of Ω (that is, f (θ, 2) = −2 and f (θ, −2) = 2). As the connected set A we take the one given by Proposition 3.12 (and Definition 3.9). 40 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES To end the proof of the theorem we need to show that T (A) = A, since this already implies that there does not exist any T -invariant curve. To see it, assume by way of contradiction that there exists an invariant curve and denote its graph by B. Since B is the graph of a (continuous) curve, it is compact and connected. On the other hand, let Ω+ and Ω− be the two connected components of Ω\A from the proof of Proposition 3.12. The facts that T (A) = A, f (θ, ·) is decreasing for every θ ∈ S1 , and T permutes the upper and lower circles of Ω imply that T (Ω+ ) = Ω− and T (Ω− ) = Ω+ . Hence, by the invariance of B, B * Ω+ and B * Ω− . The connectivity of A and B imply that there exists (θ, x) ∈ A ∩ B. Consequently, B = {T n (θ, x) : n ∈ Z+ } ⊂ A; a contradiction because A does not contain any arc of a curve. So, only it remains to prove that T (A) = A. By using Proposition 7.3(c) and i∗ (i+1)∗ Lemma 3.11(b) we get that Tm A = A , and Tk Ai∗ = Tm Ai∗ for every k, m ∈ Z , k ≥ m and i ∈ Dm . Consequently, by the definition of the map T we +  have, T (Ai ) = A(i+1) for every i ∈ Z or, equivalently, T AO (ω) = AO (ω) . ∗ ∗ ∗ ∗ Now we consider Aθ with θ ∈ S1 \O∗ (ω). Since O∗ (ω) is dense in S1 , there exists a O ∗ (ω) sequence {(θn , xn )}∞ n=0 ⊂ A such that limn→∞ θn = θ. By the compacity of A we can assume (by taking a convergent subsequence, if necessary) that {(θn , xn )}∞n=0 is convergent to a point (θ, x) ∈ A. By Lemma 3.11(c), Aθ = (θ, x) (and x = γ(θ)). On the other hand, by the part of the statement already proven, T (θn , xn ) ∈ A for every n. Hence, by the continuity of T and the compacity of A, T (θ, x) = (Rω (θ), f (θ, x)) = lim T (θn , xn ) ∈ ARω (θ) . n→∞ Since θ ∈ / O∗ (ω) we have that Rω (θ) ∈ / O∗ (ω) and, again by Lemma 3.11(c), ARω (θ) consists of a unique point.  Hence, T (Aθ ) = ARω (θ) for every θ ∈ S1 \O∗ (ω).   S1 \O ∗ (ω)  1 = A S \O (ω) . This ends the proof of the theorem. ∗ Equivalently, T A  8. Proof of Proposition 7.3 in the case m = 0 This section is devoted to prove Proposition 7.3 for m = 0; that is, to study the map T0 . It is the first technical counterpart of Section 7. To prove Proposition 7.3 for T0 we will need some more notation and a technical lemma. Given a skew product F (θ, x) = (Rω (θ), ζ(θ, x) from Ω = S1 × I to itself we define the fibre map function of F, fib(F ) : S1 −→ C(I, I) by fib(F )(θ) := ζ(θ, ·). A simple exercise shows that F is continuous if and only if ζ(θ, ·) is continuous for every θ ∈ S1 , and fib(F ) is continuous.  Lemma 8.1. Let θ ∈ Bd Bi [i∗ ] for some i ∈ D0 . Then, mi (θ) = Mi (θ) = 0, gi (θ, mi (θ)) = γ(Rω (θ)), and f0,θ is the unique piecewise affine map with two affine pieces whose graph joins the point (−2, 2) with (0, γ(Rω (θ))), and this with the point (2, −2). Proof. By Lemma 4.1(d) and Definition 7.1, we have mi (θ) = Mi (θ). Hence, f0,θ is the piecewise affine mapwith two affine pieces whose graph joins the point (−2, 2) with mi (θ), gi,θ (mi (θ)) , and this with the point (2, −2). So, we need to show that mi (θ) = 0, and gi,θ (mi (θ)) = γ(Rω (θ)). Lemma 3.6(g) and the fact that depth i = 0, Bi [i∗ ]∩B` [`∗ ] = ∅ for every ` ∈ Z|i| , i 6= `. Consequently, by Definition 3.4(R.6), mi (θ) = Mi (θ) = a− i = 0. A SKEW-PRODUCT WITHOUT FIXED-CURVES 41 Now we show that gi,θ (mi (θ)) = γ(Rω (θ)). From the definition of the map gi (Definitions 4.2 and 4.4), Lemma 3.6(e) and Definitions 3.8 and 3.4(R.1), we get gi,θ (mi (θ)) = γ|i+1| (Rω (θ)) = γ(Rω (θ)). This ends the proof of the lemma.  Proof of Proposition 7.3 for m = 0. By Lemma 3.6(b), −1 ≤ mb (θ,0) (θ) ≤ Mb (θ,0) (θ) ≤ 1 for every θ ∈ B0 . So, T0 is well defined. (b) If θ ∈ S1 \B0 , then the statement follows directly from Definition 7.1. Now assume that θ ∈ B0 and let i = b (θ, 0) . From the definition of the maps gi,θ (Definitions 4.2 and 4.4) and Definition 7.1, it follows that f0,θ I is piecewise affine i,θ and non-increasing. On the other hand, again by Definition 7.1, f0,θ [−2,mi (θ)] and f0,θ [Mi (θ),2] are affine with negative slope and f0,θ (2) = −2 and f0,θ (−2) = 2. The fact that   −1 ≤ f0,θ Mb (θ,0) (θ) ≤ f0,θ mb (θ,0) (θ) ≤ 1 for every θ ∈ B0 follows from Definition 7.1 and Lemmas 4.3(a) and 4.5(a). This ends the proof of (b). (c) Recall that [ R (i∗ ) = {θ} × Ii,θ . θ∈Bi [i∗ ] Hence, from Definition 7.1 and the definition of Gi (Definitions 4.2 and 4.4) it follows that   Tm (θ, x) = Rω (θ), fm (θ, x) = Rω (θ), gi,θ (x) = Gi (θ, x),  ∗ (i+1)∗ for every (θ, x) ∈ R (i∗ ). Thus, T0 Ai|i| = A|i+1| from Lemmas 3.10(b), 4.3(c) and 4.5(c). On the other hand, Lemma 5.3(b) implies that i∗ ∈ B0 but i∗ ∈ / Bk for every k ∈ N. Then, we get fk,i∗ = f0,i∗ from Definition 7.2. (a) Since T0 is a skew product with base Rω we only have to prove that f0 is continuous. By Definition 7.1, for every θ ∈ S1 , the map f0,θ is continuous. So we have to prove that the map fib(T0 ) (that is, the map s 7→ f0,s ) is continuous. In the rest of the proof we will denote [ IB0 := Bi (i∗ ) ⊂ B0 . i∈D0 Clearly, since for every i ∈ Z, the maps mi and Mi are continuous on Bi [i∗ ] , it follows that the map s 7→ f0,s is continuous on IB0 . Thus, we have to see that the fibre map function is continuous at every θ ∈ S1 \IB0 ; that is, limj→∞ f0,θj = f0,θ j=1 ⊂ S converging to θ. Given α > 0, we can consider four sets for every {θj }∞ 1 associated to such a sequence: {j ∈ N : θj ∈ S1 \IB0 }, {j ∈ N : θj ∈ IB0 \Bα (θ)}, {j ∈ N : θj ∈ (θ, θ + α) ∩ IB0 } and {j ∈ N : θj ∈ (θ − α, θ) ∩ IB0 }. Observe that the second set {j ∈ N : θj ∈ IB0 \Bα(θ)} is always finite and that any of the other three sets gives rise to a subsequence of {θj }∞j=1 converging to θ, when it is infinite. Consequently, the continuity of the fibre map function s 7→ f0,s at θ is equivalent to the fact that limj→∞ f0,θj = f0,θ for every {θj }∞ j=1 converging to θ and such that, for some α > 0, {θj }∞ j=1 is contained either in S1 \IB0 , or 42 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES (θ, θ + α) ∩ IB0 , or (θ − α, θ) ∩ IB0 . We will only deal with the first two cases since the proof in the last case (for (θ − α, θ)) can be done symmetrically. j=1 ⊂ S \IB0 . Case 1: limj→∞ θj = θ and {θj }∞ 1 By Definition 7.1 and Lemma 8.1, f0,θj (respectively f0,θ ) is the unique piece- wise affine map with two affine pieces whose graph joins the point (−2, 2) with (0, γ(Rω (θj ))) (respectively (0, γ(Rω (θ)))), and this with the point (2, −2). By Lemma 5.3(c) and Definition 3.8 the function γ is continuous at Rω (θ) ∈ / O∗ (ω). Hence, limj→∞ γ(Rω (θj )) = γ(Rω (θ)) and, thus, limj→∞ f0,θj = f0,θ . j=1 ⊂ (θ, θ + α) ∩ IB0 . Case 2: limj→∞ θj = θ and {θj }∞ If there exists i ∈ D0 such that θ is the left endpoint of Bi [i∗ ] ⊂ B0 then the result follows from Definition 7.1, the continuity of the maps mi and Mi and the continuity of the maps gi (Lemmas 4.3(a) and 4.5(a)). Assume now that θ is not the left endpoint of Bi (i∗ ) for every i ∈ D0 . For every j ∈ N we set ij := b (θj , 0) ∈ D0 (that is, θj ∈ Bij ((ij )∗ )). We claim that limj→∞ |ij | = ∞ and consequently, by Definition 3.4(R.1), −n|i −n|i | (17) lim 2 j +1| = lim 2 j = 0. j→∞ j→∞ To prove this claim, assume by way of contradiction that there exists L such that for every k ∈ N there exists jk ≥ k such that |ijk | ≤ L. Then, ∞ [ {θjk }∞ k=1 ⊂ Bij ((ijk )∗ ) k k=1 and, since {ijk : k ∈ N} is finite, it follows that there exists i ∈ {ijk : k ∈ N} ⊂ D0 and a subsequence of {θjk }∞ k=1 , that by abuse of notation will also be called {θjk }, such that {θjk }∞ ∗ k=1 ⊂ Bi (i ) . So, θ = lim θjk ∈ Bi [i∗ ] ; k→∞ a contradiction. So, the claim (and hence (17)) holds. Next we claim that the conditions (18) lim Mij (θj ) = lim mij (θj ) = 0, and j→∞ j→∞ there exists a sequence {xj }∞ j=1 with xj ∈ Iij ,θj = [mij (θj ), Mij (θj )] for (19) every j, such that lim f0,θ (xj ) = γ(Rω (θ)) j j→∞ imply lim f0,θj = f0,θ . j→∞ To prove the claim notice that, by Definition 7.1 and Lemma 8.1, f0,θ is the unique piecewise affine map with two affine pieces whose graph joins the point (−2, 2) with (0, γ(Rω (θ))), and this with the point (2, −2). On the other hand, for every j, • f0,θj [−2,m (θj )] is the affine map joining the point (−2, 2) with the point ij (mij (θ j ), gij (θj , mij (θj ))), and • f0,θj [M (θ ),2] is the affine map joining the point (Mij (θj ), gij (θj , Mij (θj ))) ij j with the point(2, −2) (see Figure 5). Moreover, from the part of the proposition already proven we know that f0,θj is non-increasing and continuous. Therefore, the claim holds provided that  lim diam f0,θj Iij ,θj = 0 j→∞ A SKEW-PRODUCT WITHOUT FIXED-CURVES 43 2 f0,θj γ(Rω (θ)) Iij +1,Rω (θj ) f0,θ −2 0 −2 2 Iij ,θj = [mij (θj ), Mij (θj )] Figure 5. A symbolic representation of the maps f0,θ and f0,θj in Case 2 of the proof of Proof of Proposition 7.3 for m = 0. The map f0,θ and the points 0 and γ(Rω (θ)) are drawn in blue. The map f0,θj and the corresponding intervals Iij ,θj and Iij +1,Rω (θj ) are drawn in red. (see again Figure 5). When θj ∈ Bαij [(ij )∗ ] \Bαij +1 ((ij )∗ ) and ij ≥ 0, by Definitions 7.1 and 4.2,      diam f0,θj Iij ,θj = diam gij ,θj Iij ,θj = diam {γij +1 (Rω (θj )} = 0. Otherwise, by Definition 7.1, and Lemmas 4.3(b) and 4.5(b),    {Rω (θj )} × f0,θj Iij ,θj = {Rω (θj )} × gij ,θj Iij ,θj = Gij R((ij )∗ )θj ⊂ R((ij + 1)∗ )Rω (θj ) . So, by Remark 3.2(2),  −n|i diam f0,θj Iij ,θj ≤ diam (R((ij + 1)∗ )) ≤ 2 · 2 j +1| . In any case,  −n 0 ≤ diam f0,θj Iij ,θj ≤ 2 · 2 |ij +1| for every j ∈ N  and, by (17), limj→∞ diam f0,θj Iij ,θj = 0. This ends the proof of the claim. By the last claim, to end the proof of the proposition in the case m = 0 it is enough to show that (18–19) hold. We start by proving (18). By Lemma 8.1, mij (Bd(Bij [(ij )∗ ])) = Mij (Bd(Bij [(ij )∗ ])) = 0, and from the definition of the maps mij and Mij , Definition 3.1 (or Lemma 4.1) and Remark 3.2(2), for every s ∈ Bij ((ij )∗ ) we get − 1 ≤ mij (s) < 0 < Mij (s) ≤ 1, and (20)  −n Mij (s) − mij (s) = diam Iij ,s ≤ 2 · 2 |ij | . So, (18) holds by (17). Now we prove (19). 44 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES By (2), (3) and (9), it follows that mij (θj ) < γ|i | (θj ) < Mij (θj ) if θj 6= (ij )∗ , and j mij (θj ) < γ|i (θj ) = 0 < Mij (θj ) if θj = (ij )∗ . j |−1 Also, from Definition 7.1, the definitions of Gi and gi,θ (Definitions 4.2 and 4.4), and Lemmas 4.3(c) and 4.5(c) we get f0,θj (γ|i | (θj )) = gij ,θj (γ|i | (θj )) = γ|i (Rω (θj )) if θj 6= (ij )∗ , j j j +1| f0,θj (γij −1 (θj )) = gij ,θj (γij −1 (θj )) = γij (Rω (θj )) if θj = (ij )∗ and ij ≥ 0, and f0,θj (γ|i (θj )) = gij ,θj (γ|i (θj )) = γ|i (Rω (θj )) if θj = (ij )∗ and ij < 0. j |−1 j +1| j +2| Thus, to prove (19), we have to show that   ∗ limj→∞ γ|ij +1| (Rω (θj )) = γ(Rω (θ)) if θj 6= (ij ) ,  (21) limj→∞ γij (Rω (θj )) = γ(Rω (θ)) if θj = (ij )∗ and ij ≥ 0, and   limj→∞ γ (Rω (θj )) = γ(Rω (θ)) if θj = (ij )∗ and ij < 0 |i +2| j (that is, we take xj := γ|i | (θj ) if θj 6= (ij )∗ , xj := γij −1 (θj ) if θj = (ij )∗ and ij ≥ 0, j and xj := γ|i |−1 (θj ) if θj = (ij )∗ and ij < 0). j Let ε > 0. By Lemma 5.3(c) and Definition 3.4(R.1) we have that θ ∈ / O∗ (ω) and, hence, Rω (θ) ∈ / O (ω). By the continuity of γ on S \O (ω) and the fact that ∗ 1 ∗ limi→∞ γi = γ, there exist δ > 0 and L ∈ N such that b < ε/2 for every θb ∈ Bδ (Rω (θ)) \O∗ (ω), and γ(Rω (θ)) − γ(θ) d∞ (γ, γi ) < ε/2 for every i ≥ L. Then, since limj→∞ θj = θ and limj→∞ |ij | = ∞, there exists N ∈ N such that |θ − θj | < δ/2, and |ij | ≥ L + 2 for every j ≥ N. First we will show that γ(Rω (θ)) − γ|i +1| (Rω (θj )) ≤ ε j for every j ≥ N such that θj 6= (ij )∗ . To see it observe that, by Definition 3.4(R.2) and Remark 3.5(R.2), θj , Rω (θj ) ∈ / Z|i∗j +1| whenever θj 6= (ij )∗ . Thus, γ|i +1| is j continuous at Rω (θj ) by Lemma 3.6(a).  ∗ Also, there exists a sequence {θbj` }∞ ∗ `=1 ⊂ Bδ/2(θj ) ∩ Bij ((ij ) ) \O (ω) converg- ing to θj , because S1 \O∗ (ω) is dense in S1 . Clearly, for every j ≥ N, we have {Rω (θbj` )}∞ ∗ b `=1 ⊂ Bδ (Rω (θ)) \O (ω) and lim`→∞ Rω (θj` ) = Rω (θj ). Moreover, since {Rω (θj` )}`=1 ⊂ S \O (ω) ⊂ S \Z|ij +1| , γ|i +1| is defined for every Rω (θbj` ). Then, b ∞ 1 ∗ 1 ∗ j for every j ≥ N and ` ∈ N, we have γ(Rω (θ)) − γ|ij +1| (Rω (θbj` )) ≤ γ(Rω (θ)) − γ(Rω (θbj` )) + γ(Rω (θbj` )) − γ|ij +1| (Rω (θbj` ))   < 2ε + d∞ γ, γ|i +1| < ε. j Consequently, b γ(Rω (θ)) − γ|i j +1| (R (θ ω j )) = lim γ(Rω (θ)) − γ |ij +1| (Rω j` ≤ ε (θ )) `→∞ A SKEW-PRODUCT WITHOUT FIXED-CURVES 45 This ends the proof of the first equality of (21). The second and third equalities of (21) follow as above by replacing γ|i +1| by γij (respectively γ|i +2| ), and noting that j j ( ∗ ∗ ((ij + 1)) ∈ / Zij if ij ≥ 0, and Rω (θj ) = Rω ((ij )∗ ) = ((−(|ij | − 1)))∗ ∈/ Z|i∗j |−2 if ij < 0. This ends the proof of the continuity of T0 , and the proposition for the case m = 0.  9. Proof of Proposition 7.3 for m > 0 This section is the second technical counterpart of Section 7 and is devoted to prove Proposition 7.3 for every map Tm with m > 0. To do this we will need some more technical results. Also we will use the notion of fibre map function introduced in the previous section. The next two lemmas establish some basic properties of the maps Tm V and m clarify some aspects of Definition 7.2. Lemma 9.1. For every m ∈ N and for every θ ∈ Bm , fm,θ I = gi,θ I , i,θ i,θ where i = b (θ, m) . Moreover, assume that θ ∈ WBm \WIBm . Then,    gi,θ (x) if x ∈ Ii,θ ,   2−g (mi (θ)) i,θ fm,θ (x) = 2−fm−1,θ (mi (θ)) (fm−1,θ (x) − 2) + 2 if x ∈ [−2, mi (θ)],     2+g i,θ (M i (θ)) 2+fm−1,θ (Mi (θ)) (fm−1,θ (x) + 2) − 2 if x ∈ [Mi (θ), 2]. Proof. We start by proving the first statement. When θ ∈ Bm there is nothing to prove. So, assume that θ ∈ Bm \Bm . By Definition 6.2, θ ∈ WBm , i < 0 and θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) . By Lemma 6.3(b), Ii,θ = {γ|i| (θ)} ⊂ IWm,θ . Consequently, by Definition 7.2 and the definition of the maps gi,θ for i < 0 (Defi- nition 4.4 — notice that Ii,θ ⊂ R (i∗ ) by definition),   fm,θ γ|i| (θ) = γ|i+1| (Rω (θ)) = gi,θ γ|i| (θ) . So, the first statement holds. Now we prove the second one. By Lemma 6.3(b), Ii,θ = {mi (θ)} = {Mi (θ)} = {γ|i| (θ)} = {λm (θ)} = {τm (θ)} = IWm,θ . Thus, by the part already proven, the formulas   gi,θ (x) if x ∈ Ii,θ ,   2−g (mi (θ)) i,θ 2−fm−1,θ (mi (θ)) (fm−1,θ (x) − 2) + 2 if x ∈ [−2, mi (θ)],     2+gi,θ (Mi (θ)) (f 2+fm−1,θ (Mi (θ)) m−1,θ (x) + 2) − 2 if x ∈ [Mi (θ), 2], and    γ|i+1| (Rω (θ)) if x ∈ IWm,θ ,   2−γ (Rω (θ)) |i+1| 2−fm−1,θ (λm (θ)) (fm−1,θ (x) − 2) + 2 if x ∈ [−2, λm (θ)],     2+γ|i+1| (Rω (θ)) 2+fm−1,θ (τm (θ)) (fm−1,θ (x) + 2) − 2 if x ∈ [τm (θ), 2], coincide.  46 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES Lemma 9.2. The following statements hold for every m ∈ N and i ∈ Dm : (a) The map Tm V is well defined and continuous. i∗ (b) For every θ ∈ Bi [i∗ ] , (b.i) fm,θ (2) = −2 and fm,θ (−2) = 2, (b.ii) fm,θ is piecewise affine and non-increasing, and (b.iii) −1 ≤ fm,θ (Mi (θ))  ≤∗fm,θ (mi (θ)) ≤ 1. (i+1)∗ (c) Tm ∗ = Gi and Tm A i =A . R (i ) |i| |i+1| Proof. Clearly, Tm V is well defined and continuous if and only if so is fm V . i∗ i∗ We will prove by induction on m ∈ Z+ that, (a), (b) and (b.iv) fm,θ [−2,−1] and fm,θ [1,2] are affine, fm,θ (−1) < 2 and fm,θ (1) > −2 hold for every θ ∈ Bi [i∗ ] . First we will show that (a), (b) and (b.iv) hold for m = 0 and i ∈ D0 (we are including the map f0 studied earlier to correctly start the induction process). By Proposition 7.3(a,b) for m = 0 we have that T0 V is well defined and continuous i∗ and (b) holds. By Definition 7.1, we also know that fm,θ and fm,θ [−2,mi (θ)] [Mi (θ),2] are affine. Then, (b.iv) follows from −1 ≤ mi (θ) ≤ Mi (θ) ≤ 1 (see Lemma 4.1(a)) and (b.iii). Assume now that (a), (b) and (b.iv) hold for some m − 1 ∈ Z+ and prove it for m and i ∈ Dm . By Lemma 5.2(a), θ ∈ Bi [i∗ ] Bk [k ∗ ] for some k ∈ Dm−1 . Consequently, Vi∗ ⊂ Vk∗ and fm−1 V is well defined and continuous. i∗ By Lemma 4.1(a) and Definition 6.2, −1 ≤ mi (θ) ≤ Mi (θ) ≤ 1 for θ ∈ Bi [i∗ ], and (22) −1 ≤ λm (θ) ≤ τm (θ) ≤ 1 for θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) ⊂ WBm (i < 0). Consequently, by (b.ii) and (b.iv) for m − 1, −2 < fm−1,θ (1) ≤ fm−1,θ (Mi (θ)) ≤ fm−1,θ (mi (θ)) ≤ fm,θ (−1) < 2 for every θ ∈ Bi [i∗ ], and −2 < fm−1,θ (1) ≤ fm−1,θ (τm (θ)) ≤ fm−1,θ (λm (θ)) ≤ fm,θ (−1) < 2 for θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) ⊂ WBm when i < 0. On the other hand, as it was observed in Definition 7.2, fm,θ is defined in two different ways when θ ∈ WBm ∩ Bm . In such a case, by Lemmas 6.3(a,b) and 9.1, θ∈/ WIBm and both definitions for fm,θ coincide. Hence, fm V is well defined. i∗ Now we prove that fm is continuous by using the continuity of fm−1 . Vi∗ Vi∗ Since Bα|i| [i∗ ] ⊂ Bm , by Definition 7.2, the continuity of the maps mi and Mi (see Lemma 4.1(b)), and the continuity of the maps gi (Lemmas 4.3(a) and 4.5 (a)), fm B [i∗ ] is continuous. Now we assume that i < 0 and we study the α|i| continuity of fm U on a connected component U of Bi [i∗ ] \Bα|i| (i∗ ) . Observe that, by Definition 6.2 and Lemma 5.2(b), U is a connected component of WBm . Then, again by Definition 7.2, the continuity of the maps λm U and τm U (Lemma 6.4), and the continuity of the map γ|i| U (Lemma 3.6(a) and Definition 3.4(R.2) and Remark 3.5(R.2)), fm U is continuous. Therefore, fm V is continuous because it   i∗ is well defined on  Bi [i∗ ] \Bα|i| (i∗ ) ∩ Bα|i| [i∗ ] . Let θ ∈ Bα|i| [i∗ ] ⊂ Bm . By Definition 7.2, and the definition of the maps gi,θ (Definitions 4.2 and 4.4), fm,θ I is piecewise affine and non-increasing. So, i,θ by Lemma 9.1 for m − 1 and Definition 7.2, fm,θ (2) = −2, fm,θ (−2) = 2, and A SKEW-PRODUCT WITHOUT FIXED-CURVES 47 fm,θ [−2,m (θ)] and fm,θ [M (θ),2] are affine transformations of the map fm−1,θ with i i positive slope. Hence, (b.i,ii) hold for fm,θ in this case. Moreover, (b.iv) is verified by (22) and (b.iv) for m − 1. Consider θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) ⊂ WBm . Again by Definition 7.2, fm,θ IW is m,θ constant. Then, (b.i,ii) and (b.iv) hold for fm,θ as above by replacing mi (θ) and Mi (θ) by λm (θ) and τm (θ), respectively. By (b.ii) and (22) we have fm,θ (Mi (θ)) ≤ fm,θ (mi (θ)) . Hence, (b.iii) follows from Lemma 9.1, Definition 7.2, Lemmas 4.3(b) and 3.10(c), Definition 3.4(R.2) and Remark 3.5(R.2), Lemma 4.5(b) and Lemma 3.6(b). (c) In a similar way to the proof of Proposition 7.3 for the case m = 0, [ R (i∗ ) = {θ} × Ii,θ ⊂ Vi∗ ⊂ Vm θ∈Bi [i∗ ] and, by Definition 7.2, Lemma 9.1 and the definition of Gi (Definitions 4.2 and 4.4) it follows that   Tm (θ, x) = Rω (θ), fm (θ, x) = Rω (θ), gi,θ (x) = Gi (θ, x),  ∗ (i+1)∗ for every (θ, x) ∈ R (i∗ ). Thus, Tm Ai|i| = A|i+1| from Lemmas 3.10(b), 4.3(c) and 4.5(c).  The next technical lemma compares the images of fm,θ and fm−1,θ on a point. It is an extension of Lemma 5.4. Lemma 9.3. Assume that Bi [i∗ ] ⊂ Bk [k ∗ ] for some i ∈ Dm , k ∈ Dm−1 and m ∈ N. Then, for every θ ∈ Bi [i∗ ] \Bα|i| (i∗ ) , mi (θ) = Mi (θ) = γi (θ) and fm,θ (mi (θ)) = gi,θ (mi (θ)) = γ|i+1| (Rω (θ)) , and fm−1,θ (mi (θ)) = gk,θ (mi (θ)) = γ|k+1| (Rω (θ)) . Proof. The fact that mi (θ) = Mi (θ) = γi (θ) follows directly from the definitions. The first equation follows from Lemma 9.1, and the definition of the map gi,θ (Definitions 4.2 and 4.4). By Lemma 5.4, Ii,θ = {mi (θ)} = {γ|k| (θ)} ⊂ Ik,θ . Moreover, as in the proof of Lemma 5.4, θ 6= k ∗ . Consequently, by Definition 7.1, Lemma 9.1, Lemmas 4.3(c) and 4.5(c) and (2) (alternatively, for the last equality check directly the proofs of the Lemmas 4.3(c) and 4.5(c)),  fm−1,θ (mi (θ)) = gk,θ (mi (θ)) = gk,θ γ|k| (θ) = γ|k+1| (Rω (θ)) .  The following lemma is the analogue of Lemma 8.1 for m ≥ 1. To state it we will use the set  EB = EB × I ⊂ V . m m m Lemma 9.4. Tm EB = Tm−1 EB for every m ∈ N. Equivalently, fm,θ = m m fm−1,θ for every m ∈ N and θ ∈ EBm . Proof. Fix m ∈ N and θ ∈ EBm ⊂ Bm . By Lemma 5.2(a,b), there exist i ∈ Dm and k ∈ Dm−1 such that θ ∈ Bd (Bi [i∗ ]) ⊂ Bi [i∗ ] Bk [k ∗ ] . So, we are in the assumptions of Lemmas 5.4 and 9.3 and, hence, Ii,θ = {mi (θ)} = {γ|i| (θ)} = {γ|k| (θ)} ⊂ Ik,θ , fm,θ (mi (θ)) = gi,θ (mi (θ)) = γ|i+1| (Rω (θ)) , and fm−1,θ (mi (θ)) = gk,θ (mi (θ)) = γ|k+1| (Rω (θ)) . 48 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES Thus, if i ≥ 0, θ ∈ Bm and, by Definition 7.2 and Lemma 9.2(a), to prove that fm,θ = fm−1,θ we only have to show that gi,θ (mi (θ)) = γ|i+1| (Rω (θ)) = γ|k+1| (Rω (θ)) = fm−1,θ (mi (θ)) . When i < 0, θ ∈ WBm ∩ EBm and, by Lemma 6.3(a), θ ∈ / WIBm . Then, by Lemma 9.1, we get again that gi,θ (mi (θ)) = γ|i+1| (Rω (θ)) = γ|k+1| (Rω (θ)) = fm−1,θ (mi (θ)) . implies fm,θ = fm−1,θ . If |k + 1| = |i + 1| there is nothing to prove. So, by Lemma 5.4, we can assume that |k + 1| < |i + 1| and we have γ|k+1| (Rω (θ)) = γ|k+1|+1 (Rω (θ)) = · · · = γ|i+1|−1 (Rω (θ)) . Hence, we have to show that γ|i+1| (Rω (θ)) = γ|i+1|−1 (Rω (θ)) . If i ≥ 0 we get γ|i+1| (Rω (θ)) = γi+1 (Rω (θ)) = γi (Rω (θ)) = γ|i+1|−1 (Rω (θ))  by Lemma 3.6(e). Otherwise we have i < 0, θ ∈ Bd (Bi [i∗ ]) = Bd Bα|i+1| [i∗ ] and,  consequently, Rω (θ) ∈ Bd Bα|i+1| [(i + 1)∗ ] . Again by Lemma 3.6(e) for j = |i + 1|, γ|i+1| (Rω (θ)) = γ|i+1|−1 (Rω (θ)) . This ends the proof of the lemma.  Now we aim at computing two different kind of upper bounds for kfm,θ − fm−1,θ k (Lemma 9.6 and Proposition 7.4). This will be a key tool in the proof of Propo- sitions 7.3 for m > 0 and 7.4. The next two lemmas and remark will be useful to automate and simplify the proofs of these two results. Lemma 9.5.    fm,θ − f when θ ∈ Bm \WIBm , and  Ib (θ,m),θ m−1,θ I b (θ,m),θ kfm,θ − fm−1,θ k =    fm,θ − fm−1,θ IW when θ ∈ WIBm , IW m,θ m,θ for every m ≥ 2 and θ ∈ Bm . Proof. Set i = b (θ, m) ∈ Dm , so that θ ∈ Bi [i∗ ]. When θ ∈ Bm \WIBm = Bm ∪ WBm \WIBm , by Definition 7.2 and Lemma 9.1, it is enough to show that |fm,θ (x) − fm−1,θ (x)| ≤ |fm,θ (mi (θ)) − fm−1,θ (mi (θ))| for every x ∈ [−2, mi (θ)], and |fm,θ (x) − fm−1,θ (x)| ≤ |fm,θ (Mi (θ)) − fm−1,θ (Mi (θ))| for every x ∈ [Mi (θ), 2]. We will prove the first statement. The second one follows similarly. Definition 7.2 and Lemma 9.1 give 2 − gi,θ (mi (θ)) fm,θ (x) − fm−1,θ (x) = (fm−1,θ (x) − 2) + 2 − fm−1,θ (x) 2 − fm−1,θ (mi (θ)) 2 − fm,θ (mi (θ)) = (fm−1,θ (x) − 2) − (fm−1,θ (x) − 2) 2 − fm−1,θ (mi (θ))   2 − fm,θ (mi (θ)) = (fm−1,θ (x) − 2) −1 2 − fm−1,θ (mi (θ)) fm,θ (mi (θ)) − fm−1,θ (mi (θ)) = (2 − fm−1,θ (x)) . 2 − fm−1,θ (mi (θ)) A SKEW-PRODUCT WITHOUT FIXED-CURVES 49 By Lemma 9.2(b), 2 ≥ fm−1,θ (x) ≥ fm−1,θ (mi (θ)) and 1 ≥ fm−1,θ (mi (θ)) . Hence, |fm,θ (mi (θ)) − fm−1,θ (mi (θ))| |fm,θ (x) − fm−1,θ (x)| = (2 − fm−1,θ (x)) 2 − fm−1,θ (mi (θ)) ≤ |fm,θ (mi (θ)) − fm−1,θ (mi (θ))| . Now assume that θ ∈ WIBm ⊂ WBm . By Definition 7.2 it is enough to show that |fm,θ (x) − fm−1,θ (x)| ≤ |fm,θ (λm (θ)) − fm−1,θ (λm (θ))| for every x ∈ [−2, λm (θ)], and |fm,θ (x) − fm−1,θ (x)| ≤ |fm,θ (τm (θ)) − fm−1,θ (τm (θ))| for every x ∈ [τm (θ), 2]. As before, we will prove the first statement. The second one follows similarly. We have fm,θ (λm (θ)) − fm−1,θ (λm (θ)) fm,θ (x) − fm−1,θ (x) = (2 − fm−1,θ (x)) . 2 − fm−1,θ (λm (θ)) By Lemma 9.2(b), 2 ≥ fm−1,θ (x) ≥ fm−1,θ (λm (θ)) and hence, |fm,θ (x) − fm−1,θ (x)| ≤ |fm,θ (mi (θ)) − fm−1,θ (mi (θ))| provided that 2−fm−1,θ (λm (θ)) 6= 0. Assume by way of contradiction that we have fm−1,θ (λm (θ)) = 2. Then, by Definition 6.2 and Lemma 9.2(b), −1 ≤ λm (θ) and 2 ≥ fm−1,θ (−1) ≥ fm−1,θ (λm (θ)) = 2; which contradicts statement (b.iv) from the proof of Lemma 9.2.  Next we compute an upper bound for kfm,θ − fm−1,θ k for every θ ∈ Bi [i∗ ] and i ∈ Dm such that diam(Bi [i∗ ]) is small enough. Lemma 9.6. Assume that Tm−1 is continuous for some m ≥ 2 and let ε be positive. Then, there exist %m (ε) ∈ N such that kfm,θ − fm−1,θ k ≤ ε for every θ ∈ Bi [i∗ ] and i ∈ Dm (that is, Bi [i∗ ] ⊂ Bm ) such that |i| ≥ %m (ε). Proof. Since Tm−1 is uniformly continuous, there exists δm−1 = δm−1 (ε) > 0 such that dΩ (Tm−1 (θ, x), Tm−1 (ν, y)) < ε provided that dΩ ((θ, x), (ν, y)) < δm−1 . We choose %m = %m (ε) ∈ N such that 3 · 2−%m < min{δm−1 (ε/2), ε/2}. Assume that i ∈ Dm verifies |i| ≥ %m (ε) and let (θ, x) ∈ Vi∗ = Bi [i∗ ] × I. When θ ∈ Bi [i∗ ] \WIBm we can use Lemma 9.5 with Ii,θ to compute kfm,θ − fm−1,θ k . We have to show that |fm,θ (x) − fm−1,θ (x)| < ε for every x ∈ Ii,θ . Let ν ∈ Bd (Bi [i∗ ]) ⊂ EBm . We have (θ, x), (ν, mi (ν)) ∈ R (i∗ ) and, by Lem- mas 9.2(c) and 3.6(f), dΩ (Tm (θ, x), Tm (ν, mi (ν))) = dΩ (Gi (θ, x), Gi (ν, mi (ν)))  ≤ diam Gi R (i∗ ) , and  dΩ ((θ, x), (ν, mi (ν)) ≤ diam R (i∗ ) ≤ 2 · 2−|i| < 3 · 2−%m < δm−1 (ε/2). Thus, dΩ (Tm−1 (θ, x), Tm−1 (ν, mi (ν)) < ε/2. 50 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES Consequently, by Lemma 9.4, |fm,θ (x) − fm−1,θ (x)| = dΩ (Tm (θ, x), Tm−1 (θ, x)) ≤ dΩ (Tm (θ, x), Tm−1 (ν, mi (ν))) + dΩ (Tm−1 (ν, mi (ν)), Tm−1 (θ, x)) < dΩ (Tm (θ, x), Tm (ν, mi (ν))) + ε/2  < diam Gi R (i∗ ) + ε/2. Now we look at the size of Gi (R (i∗ )) . When i < 0, from Lemmas 4.5(b) and 3.6(f), we obtain  (23) diam Gi R (i∗ ) ≤ diam (R((i + 1)∗ )) ≤ 2−(|i|−1) < 2 · 2−|i| . When i ≥ 0, from Lemma 4.3(b) we get  (Bα [(i+1)∗ ]\Bα ((i+1)∗ )) Gi R (i∗ ) = Gi (R(i∗ )) ⊂ R((i + 1)∗ ) ∪ Ai+1 i i+1 . Moreover, as in the proof of Lemma 3.6(f) for ` < 0, the set (Bα [(i+1)∗ ]\Bα ((i+1)∗ )) R((i + 1)∗ ) ∪ Ai+1 i i+1 is connected. So, by Lemma 3.6(f),    (Bα [(i+1)∗ ]\Bα ((i+1)∗ )) ∗ ∗ i i+1 diam Gi R (i ) ≤ diam R((i + 1) ) ∪ Ai+1   (Bα [(i+1)∗ ]\Bα ((i+1)∗ )) ∗ i i+1 ≤ diam (R((i + 1) )) + diam Ai+1   (Bα [(i+1)∗ ]\Bα ((i+1)∗ )) −(i+1) i i+1 ≤2 + diam Ai+1 . As noticed earlier, Bαi [(i + 1)∗ ] \Bαi+1 ((i + 1)∗ ) is disjoint from Bαi+1 ((i + 1)∗ ) ∪ B−(i+1) [(−(i + 1))∗ ] ∪ Zi+1 ∗ by Definition 3.4(R.2) and Remark 3.5(R.2). So, by Lemma 3.10(c), Definition 3.4 and Lemma 3.6(a), Aνi+1 = {(ν, γi+1 (ν)} = {(ν, γi (ν)}   ∈ {ν} × γi ((i + 1)∗ ) − 2−ni , γi ((i + 1)∗ ) + 2−ni . for every ν ∈ Bαi [(i + 1)∗ ] \Bαi+1 ((i + 1)∗ ) . On the other hand, γi ((i + 1)∗ ) ∈ Ii+1,(i+1)∗ by Lemma 3.6(c). Hence, by Remark 3.2(2), Definition 3.4(R.1) and Remark 3.5(R.1),   (Bα [(i+1)∗ ]\Bα ((i+1)∗ )) diam Ai+1 i i+1   ≤ max diam Bαi [(i + 1)∗ ] \Bαi+1 ((i + 1)∗ ) , 2 · (2−ni + 2−ni+1 )  ≤ 2 · max αi , 2−ni + 2−ni+1 = 2 · (2−ni + 2−ni+1 ) < 4 · 2−ni ≤ 2 · 2−i . Summarizing, when i ≥ 0,  diam Gi R (i∗ ) ≤ 2−(i+1) + 2 · 2−i < 3 · 2−i and, from (23),  diam Gi R (i∗ ) < 3 · 2−|i| ≤ 3 · 2−%m < ε/2 for every i ∈ Z+ . Thus, for every x ∈ Ii,θ ,  |fm,θ (x) − fm−1,θ (x)| < diam Gi R (i∗ ) + ε/2 < ε. A SKEW-PRODUCT WITHOUT FIXED-CURVES 51 Now assume that θ ∈ Bi [i∗ ] ∩ WIBm . We can use Lemma 9.5 with IWm,θ to compute kfm,θ − fm−1,θ k . We have to show that |fm,θ (x) − fm−1,θ (x)| < ε for every x ∈ IWm,θ . Since θ ∈ WIBm , by Definition 6.2 and Lemma 6.3(b), i < 0, θ ∈ WBm and  Ii,θ = γ|i| (θ) ⊂ IWm,θ = I`,θ 3 x with ` = b (θ, led (θ, m)) ∈ WFDm . In this case we will consider the points (θ, x) ∈ R(`∗ ) and (ν, mi (ν)), (θ, γ|i| (θ)) ∈ R (i∗ ) with ν ∈ Bd (Bi [i∗ ]) ⊂ EBm . By Lemma 6.1(b), Remark 3.2(2) and Lemma 3.6(f), |i| < |`| and dΩ ((θ, x), (ν, mi (ν)) ≤ dΩ ((θ, x), (θ, γ|i| (θ)) + dΩ ((θ, γ|i| (θ)), (ν, mi (ν))  ≤ x − γ|i| (θ) + diam R (i∗ )  ≤ diam (R(`∗ )) + diam R (i∗ ) ≤ 2−|`| + 2 · 2−|i| < 3 · 2−|i| ≤ 3 · 2−%m < δm−1 (ε/2). Thus, dΩ (Tm−1 (θ, x), Tm−1 (ν, mi (ν)) < ε/2. On the other hand, by Lemma 9.2(c), Definition 7.2 and (23), dΩ (Tm (θ, x), Tm (ν, mi (ν))) ≤ dΩ (Tm (θ, x), Tm (θ, γ|i| (θ))) + dΩ (Tm (θ, γ|i| (θ)), Tm (ν, mi (ν))) ≤ fm,θ (x) − fm,θ (γ|i| (θ)) + dΩ (Gi (θ, γ|i| (θ)), Gi (ν, mi (ν)))  = dΩ (Gi (θ, x), Gi (ν, mi (ν))) ≤ diam Gi R (i∗ ) < 2 · 2−|i| ≤ 3 · 2−%m < ε/2. So, in a similar way as before, Lemma 9.4 gives |fm,θ (x) − fm−1,θ (x)| = dΩ (Tm (θ, x), Tm−1 (θ, x)) ≤ dΩ (Tm (θ, x), Tm−1 (ν, mi (ν))) + dΩ (Tm−1 (ν, mi (ν)), Tm−1 (θ, x)) < ε.  Proof of Proposition 7.3 for m > 0. (a) We start by proving by induction on m that Tm is continuous for every m ∈ Z+ . By Proposition 7.3(a) for m = 0, T0 is continuous. So, we may assume that Tm−1 is continuous for some m ∈ N and prove that Tm is continuous. Let ε > 0 be fixed but arbitrary, and let (θ, x), (ν, y) ∈ Ω. We have to show that there exists δ(ε) > 0 such that dΩ (Tm (θ, x), Tm (ν, y)) < ε when dΩ ((θ, x), (ν, y)) < δ. We start by defining δ(ε). To this end we need to introduce some more notation and establish some facts about the maps Tm and Tm−1 . Since Tm−1 is uniformly continuous, we know that there exists δm−1 = δm−1 (ε) > 0 such that dΩ (Tm−1 (θ, x), Tm−1 (ν, y)) < ε (24) provided that dΩ ((θ, x), (ν, y)) < δm−1 . On the other hand, Lemma 9.2(a) tells us that Tm is uniformly continuous Vi∗ for every i ∈ Dm . So, for every i ∈ Dm , there exists δm,i = δm,i (ε) > 0 such that dΩ (Tm (θ, x), Tm (ν, y)) < ε for (25) every (θ, x), (ν, y) ∈ Vi∗ ⊂ Vm verifying dΩ ((θ, x), (ν, y)) < δm,i (ε). 52 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES Then, by using the numbers δm−1 (ε/7) given by (24), δm,i (ε/7) given by (25) and %m (ε/7) given by Lemma 9.6, we set  δ = δ(ε) := min δm−1 (ε/7), min{δm,i (ε/7) : i ∈ Dm ∩ Z%m (ε/7) } . Clearly, δ > 0 because the set Dm ∩ Z%m (ε/7) is finite. Now we will show that if dΩ ((θ, x), (ν, y)) < δ, then dΩ (Tm (θ, x), Tm (ν, y)) < ε. Assume first that (θ, x), (ν, y) ∈ V`∗ for some ` ∈ Dm ∩ Z%m (ε/7) . We have dΩ ((θ, x), (ν, y)) < δ ≤ min{δm,i (ε/7) : i ∈ Dm ∩ Z%m (ε/7) } ≤ δm,` (ε/7). Hence, by (25), dΩ (Tm (θ, x), Tm (ν, y)) < ε/7 < ε. Next we assume that (θ, x), (ν, y) ∈ V`∗ for some ` ∈ Dm such that |`| > %m (ε/7) (in particular, θ, ν ∈ B` [`∗ ]). In this situation we have dΩ ((θ, x), (ν, y)) < δ ≤ δm−1 (ε/7) and, by (24) and Lemma 9.6, dΩ (Tm (θ, x), Tm (ν, y)) ≤ dΩ (Tm (θ, x), Tm−1 (θ, x)) + dΩ (Tm−1 (θ, x), Tm−1 (ν, y)) + dΩ (Tm−1 (ν, y), Tm (ν, y)) = |fm,θ (x) − fm−1,θ (x)| + dΩ (Tm−1 (θ, x), Tm−1 (ν, y)) + |fm,ν (y) − fm−1,ν (y)| ≤ kfm,θ − fm−1,θ k + dΩ (Tm−1 (θ, x), Tm−1 (ν, y)) + kfm,ν − fm−1,ν k 3 < 7ε < ε. In summary, we have proved that dΩ (Tm (θ, x), Tm (ν, y)) < 73 ε when dΩ ((θ, x), (ν, y)) < δ and (θ, x), (ν, y) ∈ V`∗ for some ` ∈ Dm . Next we assume that (θ, x), (ν, y) ∈ Vm but (θ, x), (ν, y) ∈ / V`∗ for every ` ∈ Dm . By Lemma 5.2(a,b), there exist i = b (θ, m) , k = b (ν, m) ∈ Dm , i 6= k, such that θ ∈ Bi [i∗ ] , (θ, x) ∈ Vi∗ , ν ∈ Bk [k ∗ ] and (ν, y) ∈ Vk∗ . Then, there exist   θe ∈ A ∩ Bd Bi [i∗ ] ⊂ EBm and νe ∈ A ∩ Bd Bk [k ∗ ] ⊂ EBm , where A denotes the closed arc of S1 such that diam(A) = dS1 (θ, ν) and Bd(A) = {θ, ν}.   e x ∈ V ∗ , (ν, y), νe, y ∈ V ∗ and, by the previous case, Clearly we have, (θ, x), θ, i k    e x = d 1 θ, θe ≤ d 1 (θ, ν) ≤ d ((θ, x), (ν, y)) < δ, dΩ (θ, x), θ, S S Ω   3 ex < ε dΩ Tm (θ, x), Tm θ, 7   dΩ (ν, y), νe, y = dS1 ν, νe ≤ dS1 (θ, ν) ≤ dΩ ((θ, x), (ν, y)) < δ, and  dΩ Tm (ν, y), Tm νe, y < 73 ε. A SKEW-PRODUCT WITHOUT FIXED-CURVES 53   On the other hand, θ,e x , νe, y ∈  EB ⊂ V ⊂ V and, by Lemma 9.4 and m m m−1 (24),    n  o  d e x , νe, y = max d 1 θ, θ, e νe , |x − y| ≤ max d 1 (θ, ν), |x − y| Ω S S = dΩ ((θ, x), (ν, y)) < δ ≤ δm,i (ε/7), and      dΩ (Tm (θ, x), Tm (ν, y)) ≤ dΩ Tm (θ, x), Tm θ, e x + d Tm θ, e x , Tm νe, y + Ω   dΩ Tm νe, y , Tm (ν, y)    < 73 ε + dΩ Tm−1 θ, e x , Tm−1 νe, y + 3 ε = ε. 7 / Vm then, by Definition 7.2 and (24) , If (θ, x), (ν, y) ∈ dΩ (Tm (θ, x), Tm (ν, y)) = dΩ (Tm−1 (θ, x), Tm−1 (ν, y)) < ε/7 < ε because dΩ ((θ, x), (ν, y)) < δ ≤ δm−1 (ε/7). Lastly, assume that (ν, y) ∈ / Vm but (θ, x) ∈ Vi∗ ⊂ Vm , for some i ∈ Dm (that is, θ ∈ Bi [i ]). In this situation, as before, there exists θe ∈ Bd (Bi [i∗ ]) ⊂ EBm such ∗  that, by Lemma 9.4 and Definition 7.2 ( θ, e x ∈  EB ⊂ V ⊂ V ), and (24), m m m−1   dΩ (θ, x), θ,e x < δ,    dΩ θ, e x , (ν, y < δ ≤ δm−1 (ε/7),   dΩ Tm (θ, x), Tm θ, e x < 3 ε, and 7      e x + d Tm θ, dΩ (Tm (θ, x), Tm (ν, y)) ≤ dΩ Tm (θ, x), Tm θ, e x , Tm (ν, y) Ω    e x , Tm−1 (ν, y) < ε. < 73 ε + dΩ Tm−1 θ, This ends the proof of the continuity of Tm and, hence, of (a). (b) When θ ∈ Bm the statement follows from Lemma 9.2(b). When θ ∈ S1 \Bm , it follows from the part already proven and the continuity of Tm . (c) The first two statements follow from Lemma 9.2(c) and statement (a). On the other hand, as in the proof of Proposition 7.3(c) for m = 0, Lemma 5.3(b) implies that i∗ ∈ Bm but i∗ ∈ / Bk for every k > m. Then, we get fk,i∗ = fm,i∗ from Definition 7.2.  10. Proof of Proposition 7.4 This section is devoted to prove Proposition 7.4. It is the third technical coun- terpart of Section 7. In contrast to Lemma 9.6 the bound given by Proposition 7.4. is valid for every θ ∈ Bm . Before starting the proof of this proposition we will state and prove a number of very simple lemmas that will help in automating the proof of Proposition 7.4. Lemma 10.1. Assume that Bi [i∗ ] ⊂ Bk [k ∗ ] for some i ∈ Dm , k ∈ Dm−1 and m ≥ 2, and assume that either i < 0 and θ ∈ Bi [i∗ ] \{i∗ } or i ≥ 0 and θ ∈ Bαi [i∗ ] \Bαi+1 (i∗ ) . Then, γ (Rω (θ)) − γ|k+1| (Rω (θ)) ≤ 2−|k| . |i+1| Proof. The lemma holds trivially when |k + 1| = |i + 1|. Thus, we may assume that |k + 1| = 6 |i + 1|. Then by Lemma 5.4, |k| < |i| , |k + 1| < |i + 1| and γ|k+1| (Rω (θ)) = γ|k+1|+1 (Rω (θ)) = · · · = γ|i+1|−1 (Rω (θ)) . 54 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES By assumption we have ( Bαi [i∗ ] \Bαi+1 (i∗ ) when i ≥ 0, and θ∈ Bi [i∗ ] \{i∗ } = Bα|i+1| [i∗ ] \{i∗ } when i < 0, and, hence, ( Bαi [(i + 1)∗ ] \Bαi+1 ((i + 1)∗ ) when i ≥ 0, and Rω (θ) ∈ Bα|i+1| [(i + 1)∗ ] \{(i + 1)∗ } when i < 0. Thus, in the case i ≥ 0 we have / Bαi+1 ((i + 1)∗ ) ∪ B−(i+1) [(−(i + 1))∗ ] ∪ Zi+1 Rω (θ) ∈ ∗ by Definition 3.4(R.2) and Remark 3.5(R.2). So, by Definition 3.4, γi+1 (Rω (θ)) = γi (Rω (θ)) = γ|k+1| (Rω (θ)) . This ends the proof of the lemma in this case. Assume now that i < 0. By Lemma 3.6(c,d,f) and Definition 3.4(R.2) and Re- mark 3.5(R.2), γ (Rω (θ)) − γ (Rω (θ)) = γ (Rω (θ)) − γ (Rω (θ)) |i+1| |k+1| |i+1| |i+1|−1 ≤ diam (R((i + 1) )) ≤ 2−|i+1| ≤ 2−|k| ∗ (observe that |i + 1| > |k + 1| ≥ |k| − 1).  Lemma 10.2. Let s, t ∈ Z, s 6= t be such that θ ∈ Bs (s∗ ) \Bα|s| (s∗ ) , and either t < 0 and θ ∈ Bα|t| (t∗ ) or t ≥ 0 and θ ∈ Bαt+1 (t∗ ) . Then, the following statements hold: (a) Rω (θ) ∈ Bα|s+1| ((s + 1)∗ ) ∩ Bα|t+1| ((t + 1)∗ ) . (b) Let u, v ∈ Z be such that {u, v} = {s, t} and |u + 1| ≤ |v + 1|. Then, Iv+1,Rω (θ) ⊂ Iu+1,Rω (θ) . (c) |x − y| ≤ 2 · 2−|u| for every x ∈ It+1,Rω (θ) and y ∈ Is+1,Rω (θ) . Proof. By assumption we have ( Bαt+1 (t∗ ) when t ≥ 0, and θ∈ Bα|t| (t∗ ) ⊂ Bt (t∗ ) = Bα|t+1| (t∗ ) when t < 0. Hence, Rω (θ) ∈ Bα|t+1| ((t + 1)∗ ) . Moreover, as in the proof of Lemma 10.1, s < 0 and Rω (θ) ∈ Bα|s+1| ((s + 1)∗ ) . This proves (a). Now we prove (b). From (a) we have Rω (θ) ∈ Bα|u+1| ((u + 1)∗ ) ∩ Bα|v+1| ((v + 1)∗ ) ⊂ Bα|u+1| ((u + 1)∗ ) ∩ Bv+1 [(v + 1)∗ ] . Moreover, s 6= t implies u + 1 6= v + 1 and we have |u + 1| ≤ |v + 1| by assumption. Consequently, by Lemma 3.6(g,d) and Definition 3.4(R.2) and Remark 3.5(R.2), |u + 1| < |v + 1| and   R((v + 1)∗ ) ⊂ Int R((u + 1)∗ )\  {(u + 1)∗ } which implies (b). Thus, x, y ∈ Iu+1,Rω (θ) and, by Lemma 3.6(f), |x − y| ≤ diam (R((u + 1)∗ )) ≤ 2−|u+1| ≤ 2−(|u|−1) = 2 · 2−|u| .  A SKEW-PRODUCT WITHOUT FIXED-CURVES 55 Now we are ready to start the proof of Proposition 7.4. Proof of Proposition 7.4. We start by showing that {Tm }∞ k=0 is a Cauchy sequence, assuming that the bound (16) holds for every m ≥ 2 and θ ∈ S1 . We start by estimating d∞ (Tm , Tm+1 ) for every m ∈ N. From (16) and the definition of µm d∞ (Tm , Tm+1 ) = sup kfm,θ − fm+1,θ k ≤ 2 · sup 2−|b (θ,m)| ≤ 2 · 2−µm . θ∈S1 θ∈S1 By Lemma 5.3(a) {µm }∞ m=0is strictly increasing (and limm→∞ µm = ∞). There- fore, for every ε > 0, there exists N ≥ 2, such that 4 · 2−µm < ε for every m ≥ N. Hence, m+i−1 X m+i−1 X d∞ (Tm , Tm+i ) ≤ d∞ (T` , T`+1 ) ≤ 2 · 2−µ` `=m `=m ∞ X ≤ 2 · 2−µm 2−` = 4 · 2−µm ≤ 4 · 2−µN < ε `=0 for every m ≥ N and i ∈ N. So, {Tm }∞ k=0 is a Cauchy sequence. Now we prove (16). That is, kfm,θ − fm−1,θ k ≤ 2 · 2−|b (θ,m−1)| for every m ≥ 2 and θ ∈ S1 . From Definition  7.2 and Lemma 9.4 we know that fm,θ = fm−1,θ for every θ ∈ S1 \Bm ∪ EBm . Then, (16) holds in this case. In the rest of the involved proof we assume that θ ∈ Bm \EBm . Thus, by Lem- mas 5.2(a,b), 3.6(g) and 5.4,   θ ∈ Bi (i∗ ) ⊂ Bk (k ∗ ) \ Bd Bα|k| [k ∗ ] ∪ {k ∗ } where i = b (θ, m) ∈ Dm , k = b (θ, m − 1) ∈ Dm−1 , |k| < |i| , and |k + 1| ≤ |i + 1|. Moreover, Vi∗ ⊂ Vk∗ ⊂ Vm−1 . Consequently, by Lemma 9.2(a,b), the maps fm,θ and fm−1,θ are well defined, continuous, piecewise affine and non-increasing, and fm,θ (2) = fm−1,θ (2) = −2 and fm,θ (−2) = fm−1,θ (−2) = 2 (see Figures 6, 7 and 8 for some examples in generic cases). We split the proof into three cases according to whether θ belongs to Bi (i∗ ) \Bα|i| (i∗ ) , Bα|i| (i∗ ) ⊂ Bk (k ∗ ) \Bα|k| [k ∗ ] or Bα|i| (i∗ ) ⊂ Bα|k| (k ∗ ) . Case 1. θ ∈ Bi (i∗ ) \Bα|i| (i∗ ) . We have i < 0 because Bi (i∗ ) = Bαi (i∗ ) for i ≥ 0. Moreover, by Definition 6.2, θ ∈ WBm . To deal with this case we consider three subcases.  Subcase 1.1. θ ∈ Bi (i∗ ) \Bα|i| (i∗ ) \WIBm . By Lemmas 5.4, 9.3, 9.5 and 10.1, Ii,θ = {mi (θ)} = {γ|i| (θ)} = {γ|k| (θ)} ⊂ Ik,θ , fm,θ (mi (θ)) = γ|i+1| (Rω (θ)) , fm−1,θ (mi (θ)) = γ|k+1| (Rω (θ)) , and kfm,θ − fm−1,θ k = fm,θ I − fm−1,θ I = |fm,θ (mi (θ)) − fm−1,θ (mi (θ))| i,θ i,θ = γ|i+1| (Rω (θ)) − γ|k+1| (Rω (θ)) ≤ 2−|b (θ,m−1)| . 56 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES 2  Mk+1 Rω (θ) gk,θ Ik+1,Rω (θ) fm,θ  γ|i+1| (Rω (θ)) mk+1 Rω (θ) fm−1,θ −2 −2 λm (θ) IWm,θ τm (θ) 2 mk (θ) Ik,θ Mk (θ) Figure 6. A symbolic representation of the maps fm,θ and  fm−1,θ in Subcase 1.3 of Proposition 7.4 (θ ∈ Bi (i∗ ) \Bα|i| (i∗ ) ∩ WIBm and Bi (i∗ ) ⊂ Bα|k| (k ∗ ) \{k ∗ }). The map fm−1,θ and the corre- sponding intervals Ik,θ and Ik+1,Rω (θ) are drawn in blue. The map fm,θ , the interval IWm,θ and the point γ|i+1| (Rω (θ)) are drawn in red.  Subcase 1.2. θ ∈ Bi (i∗ ) \Bα|i| (i∗ ) ∩ WIBm and Bi (i∗ ) ⊂ Bk (k ∗ ) \Bα|k| [k ∗ ] . In this subcase, by Definition 6.2 we have θ ∈ Bk (k ∗ ) \Bα|k| [k ∗ ] ⊂ WBm−1 (recall that i < 0). Then, by Lemmas 5.4 and 6.3(b,c), Definition 7.2 and Lem- mas 9.5 and 10.1, Ii,θ = {γ|i| (θ)} = {γ|k| (θ)} ⊂ IWm,θ = IWm−1,θ , fm,θ (x) = γ|i+1| (Rω (θ)) for every x ∈ IWm,θ , fm−1,θ (x) = γ|k+1| (Rω (θ)) for every x ∈ IWm−1,θ , and kfm,θ − fm−1,θ k = f m,θ IWm,θ − f m−1,θ IW m,θ = γ|i+1| (Rω (θ)) − γ|k+1| (Rω (θ)) ≤ 2−|b (θ,m−1)| . Observe that since Bi (i∗ ) is connected and   Bi (i∗ ) ⊂ Bk (k ∗ ) \ Bd Bα|k| [k ∗ ] ∪ {k ∗ } , Bi (i∗ ) 6⊂ Bk (k ∗ ) \Bα|k| [k ∗ ] implies Bi (i∗ ) ⊂ Bα|k| (k ∗ ) \{k ∗ }.  Subcase 1.3. θ ∈ Bi (i∗ ) \Bα|i| (i∗ ) ∩ WIBm and Bi (i∗ ) ⊂ Bα|k| (k ∗ ) \{k ∗ } (see Figure 6 for a symbolic representation of this case). A SKEW-PRODUCT WITHOUT FIXED-CURVES 57 By Lemmas 5.4 and 6.3(b) and Definition 7.2, Ii,θ = {γ|i| (θ)} = {γ|k| (θ)} ⊂ IWm,θ , and fm,θ (x) = γ|i+1| (Rω (θ)) for every x ∈ IWm,θ . On the other hand, by Definition 6.2 and Lemma 6.1(a,b), θ ∈ WIBm ⊂ WDBm , and θ ∈ Bα|`| [`∗ ] ⊂ Bi (i∗ ) \Bα|i| [i∗ ] ⊂ Bα|k| (k ∗ ) \{k ∗ } with ` = b (θ, led (θ, m)) ∈ WFDm and |`|  > |i| > |k| . Then, by Lemma 3.6(g) and ∗ ∗  ∗ Definition 6.2, R(` ) ⊂ Int R(k )\ k and IWm,θ = I`,θ ⊂ Ik,θ . Moreover, since θ ∈ Bα|k| (k ∗ ) ⊂ Bm , Definition 7.2, Lemmas 4.3(b) and 4.5(b), and the definition of the maps gi,θ for i ≥ 0 (Definition 4.2) give  fm−1,θ IWm,θ ⊂ fm−1,θ (Ik,θ ) ( Ik+1,Rω (θ) if k < 0 or k ≥ 0 and θ ∈ Bαk+1 (k ∗ ) , ⊂ {γk+1 (Rω (θ))} if k ≥ 0 and θ ∈ Bαk [k ∗ ] \Bαk+1 (k ∗ ). Now, as before, we will use Lemma 9.5 to bound kfm,θ − fm−1,θ k . We start with the simplest case: k ≥ 0 and θ ∈ Bαk [k ∗ ] \Bαk+1 (k ∗ ) . By Lemma 10.1, kfm,θ − fm−1,θ k = fm,θ IW − fm−1,θ IW m,θ m,θ = γ |i+1| (Rω (θ)) − γ|k+1| (Rω (θ)) ≤ 2−|b (θ,m−1)| . Now we assume that k < 0 or k ≥ 0 and θ ∈ Bαk+1 (k ∗ ) . In this case Lemma 10.2 applies. By Lemmas 10.2, 3.6(d) and Definition 3.4(R.2) and Remark 3.5(R.2), and Lemma 9.5 we have γ|i+1| (Rω (θ)) ∈ Ii+1,Rω (θ) ⊂ Ik+1,Rω (θ) , fm−1,θ (x) ∈ Ik+1,Rω (θ) for every x ∈ IWm,θ . and kfm,θ − fm−1,θ k = sup |fm,θ (x) − fm−1,θ (x)| x∈IWm,θ = sup γ (Rω (θ)) − fm−1,θ (x) |i+1| x∈IWm,θ ≤ 2 · 2−|k| = 2 · 2−|b (θ,m−1)| . This ends the proof of the proposition in this case. Case 2. θ ∈ Bα|i| (i∗ ) ⊂ Bk (k ∗ ) \Bα|k| [k ∗ ] (see Figure 7 for a symbolic representa- tion of this case). case we will use Lemma 9.5 with Ii,θ . Thus, we need to compare the maps In this fm,θ I and fm−1,θ I . i,θ i,θ Directly from the definitions we get k < 0, Bα|i| [i∗ ] ⊂ Bm and Bα|k| [k ∗ ] ⊂ Bm−1 . Consequently, by Lemma 5.2(b) and Definition 6.2, θ ∈ Bm and θ ∈ Bm−1 \Bm−1 ⊂ WDBm−1 ⊂ WBm−1 . Moreover, led (θ, m − 1) = m, i = b (θ, m) = b (θ, led (θ, m − 1)) ∈ WFDm−1 and, by Definition 6.2, θ ∈ WIBm−1 , and IWm−1,θ = Ii,θ . 58 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES 2  Mi+1 Rω (θ) gi,θ Ii+1,Rω (θ) fm−1,θ Ik+1,Rω (θ)  mi+1 Rω (θ) γ|k+1| (Rω (θ)) fm,θ −2 −2 2 mi (θ) = λm−1 (θ) Ii,θ = IWm−1,θ Mi (θ) = τm−1 (θ) Figure 7. A symbolic representation of the maps fm,θ and fm−1,θ in Case 2 (θ ∈ Bα|i| (i∗ ) ⊂ Bk (k ∗ ) \Bα|k| [k ∗ ]) of Proposition 7.4. The map fm−1,θ and the corresponding intervals IWm−1,θ and Ik+1,Rω (θ) are drawn in blue. The map fm,θ and the corresponding intervals Ii,θ = IWm−1,θ and Ii+1,Rω (θ) are drawn in red. Furthermore, since k < 0, as in the proof of Lemma 10.1, Rω (θ) ∈ Bα|k+1| ((k + 1)∗ ) . Thus, Definition 7.2, Lemma 3.6(d) and Definition 3.4(R.2) and Remark 3.5(R.2), give fm−1,θ (x) = γ|k+1| (Rω (θ)) ∈ Ik+1,Rω (θ) for every x ∈ Ii,θ = IWm−1,θ . Now we will use Lemma 9.5 to bound the norm kfm,θ − fm−1,θ k . By Defini- tion 6.2 and Lemma 9.5, θ ∈ Bm ⊂ Bm \WIBm , and kfm,θ − fm−1,θ k = sup |fm,θ (x) − fm−1,θ (x)| x∈Ii,θ = sup fm,θ (x) − γ|k+1| (Rω (θ)) . x∈Ii,θ Next we will compute fm,θ (Ii,θ ). We start with the simplest case: i ≥ 0 and θ ∈ Bαi (i∗ ) \Bαi+1 (i∗ ) . By Definition 7.2, the definition of the maps gi,θ for i ≥ 0 (Definition 4.2) and Lemma 10.1, kfm,θ − fm−1,θ k = sup fm,θ (x) − γ|k+1| (Rω (θ)) x∈Ii,θ = γi+1 (Rω (θ)) − γ |k+1| (Rω (θ)) ≤ 2−|b (θ,m−1)| . A SKEW-PRODUCT WITHOUT FIXED-CURVES 59 2  Mk+1 Rω (θ) gk,θ Ik+1,Rω (θ) fm,θ  Mi+1 Rω (θ) gi,θ Ii+1,Rω (θ)   mk+1 Rω (θ) mi+1 Rω (θ) fm−1,θ −2 −2 mi (θ) Ii,θ Mi (θ) 2 mk (θ) Ik,θ Mk (θ) Figure 8. A symbolic representation of the maps fm,θ and fm−1,θ in Subcase 3.1 from the proof of Proposition 7.4 (θ ∈ Bα|i| (i∗ ) and Ii,θ ⊂ Ik,θ and either k < 0 or k ≥ 0 and i∗ ∈ Bαk+1 [k ∗ ]). The map fm−1,θ and the corresponding intervals Ik,θ and Ik+1,Rω (θ) are drawn in blue. The map fm,θ and the corresponding intervals Ii,θ and Ii+1,Rω (θ) are drawn in red. Assume that i < 0 or i ≥ 0 and θ ∈ Bαi+1 (i∗ ) . Then, again by Definition 7.2 and Lemmas 4.3(b), 4.5(b) and 10.2, fm,θ (x) ∈ Ii+1,Rω (θ) ⊂ Ik+1,Rω (θ) for every x ∈ Ii,θ , and kfm,θ − fm−1,θ k = sup fm,θ (x) − γ|k+1| (Rω (θ)) x∈Ii,θ ≤ 2 · 2−|k| = 2 · 2−|b (θ,m−1)| . This ends the proof of the proposition in Case 2. Case 3. θ ∈ Bα|i| (i∗ ) ⊂ Bα|k| (k ∗ ) . In this case we have Bα|i| (i∗ ) ⊂ Bm and Bα|k| (k ∗ ) ⊂ Bm−1 so that, θ ∈ Bm ∩ Bm−1 .   Moreover, by Lemma 3.6(g), R(i∗ ) ⊂ Int R(k ∗ )\  k ∗ and, hence, Ii,θ ⊂ Ik,θ . Since θ ∈ Bm , by Definition 6.2 and Lemma 9.5, θ ∈ Bm \WIBm , and kfm,θ − fm−1,θ k = fm,θ I − fm−1,θ I = sup |fm,θ (x) − fm−1,θ (x)| . i,θ i,θ x∈Ii,θ 60 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES Thus, we need to compare the maps fm,θ I and fm−1,θ I . To do this we consider i,θ i,θ two subcases. Subcase 3.1. Either k < 0 or k ≥ 0 and θ ∈ Bαk+1 (k ∗ ) (see Figure 8 for a symbolic representation of this case). In this situation we aim at proving that fm−1,θ (Ii,θ ) , fm,θ (Ii,θ ) ⊂ Ik+1,Rω (θ) . We start with fm−1,θ (Ii,θ ) . By Definition 7.2 and Lemmas 4.3(b) and 4.5(b) we obtain fm−1,θ (Ii,θ ) ⊂ fm−1,θ (Ik,θ ) = gk,θ (Ik,θ ) ⊂ Ik+1,Rω (θ) . Next we show that fm,θ (Ii,θ ) ⊂ Ik+1,Rω (θ) . Since k < 0 or k ≥ 0 and θ ∈ Bαk+1 (k ∗ ) , by Definition 3.4(R.1) we obtain (  Rω Bα|k| (k ∗ ) = Bα|k| ((k + 1)∗ ) ⊂ Bα|k+1| ((k + 1)∗ ) if k < 0, (26) Rω (θ) ∈  Rω Bαk+1 (k ∗ ) = Bαk+1 ((k + 1)∗ ) if k ≥ 0 and θ ∈ Bαk+1 (k ∗ ). Assume that i < 0 or i ≥ 0 and θ ∈ Bαi+1 (i∗ ) . By (26) with k replaced by i, Rω (θ) ∈ Bα|i+1| ((i + 1)∗ ) ∩ Bαk+1 ((k + 1)∗ ) ⊂ Bi+1 [(i + 1)∗ ] ∩ Bk+1 [(k + 1)∗ ] . Therefore, since |k + 1| ≤ |i + 1| and k + 1 6= i + 1, from Lemma 3.6(g) we obtain |k + 1| < |i + 1|, Bα|i+1| [(i + 1)∗ ] ⊂ Bα|k+1| ((k + 1)∗ ) \{(k + 1)∗ }, and   R((i + 1)∗ ) ⊂ Int R((k + 1)∗ )\  (k + 1)∗ . Thus, by Definition 7.2 and Lemmas 4.3(b) and 4.5(b), fm,θ (Ii,θ ) = gi,θ (Ii,θ ) ⊂ Ii+1,Rω (θ) ⊂ Ik+1,Rω (θ) . Now we will consider the case i ≥ 0 and θ ∈ Bαi (i∗ ) \Bαi+1 (i∗ ) . The fact that |k| < |i| = i implies |k + 1| ≤ |k| + 1 ≤ i. We claim that Bαi ((i + 1)∗ ) ⊂ Bα|k+1| ((k + 1)∗ ) \{(k + 1)∗ }. To prove the claim note that, by (26), Rω (θ) ∈ Rω (Bαi (i∗ )) ∩ Bα|k+1| ((k + 1)∗ ) ⊂ Bαi ((i + 1)∗ ) ∩ Bk+1 [(k + 1)∗ ] . Moreover, the interval Bαi ((i + 1)∗ ) is disjoint from Bi [i∗ ] and B−i[(−i)∗ ] by Def- inition 3.4(R.2). Thus, i 6= k + 1, −(k + 1) and, hence, |k + 1| < i (that is, k + 1 ∈ Zi−1 ). So, there exists q ∈ Zi−1 such that Bαi [(i + 1)∗ ] ∩ Bq [q ∗ ] 6= ∅ and |q| ≥ |k + 1| is maximal verifying these conditions. By Definition 3.4(R.4),   Bαi ((i + 1)∗ ) ⊂ Bq (q ∗ ) \ Bd Bα|q| [q ∗ ] ∪ {q ∗ } . So, the claim holds when q = k + 1. Assume that q 6= k + 1. Then, Rω (θ) ∈ Bαi ((i + 1)∗ ) ∩ Bα|k+1| ((k + 1)∗ ) ⊂ Bq (q ∗ ) ∩ Bα|k+1| ((k + 1)∗ ) . Hence, by Lemma 3.6(g), |q| > |k + 1| and Bαi ((i + 1)∗ ) ⊂ Bq [q ∗ ] ⊂ Bα|k+1| ((k + 1)∗ ) \{(k + 1)∗ }. This ends the proof of the claim. On the other hand, by Definition 3.4(R.2) and Remark 3.5(R.2),  Bαi [(i + 1)∗ ] \Bαi+1 ((i + 1)∗ ) ∩ Zi+1 = ∅. A SKEW-PRODUCT WITHOUT FIXED-CURVES 61 Thus, by the claim,  Rω (θ) ∈ Rω Bαi (i∗ ) \Bαi+1 (i∗ ) = Bαi ((i + 1)∗ ) \Bαi+1 ((i + 1)∗ ) ⊂ Bα|k+1| ((k + 1)∗ ) \Zi+1 . By Definition 7.2, the definition of the maps gi,θ for i ≥ 0 (Definition 4.2) and Lemma 3.6(d) (with ` = k + 1 and n = i + 1),  fm,θ (Ii,θ ) = gi,θ (Ii,θ ) = γi+1 (Rω (θ)) ⊂ Ik+1,Rω (θ) . Summarizing, we have proved that fm−1,θ (Ii,θ ) , fm,θ (Ii,θ ) ⊂ Ik+1,Rω (θ) . So, by Lemma 3.6(f) (and the fact that |k + 1| ≥ |k| − 1),  kfm,θ − fm−1,θ k = sup |fm,θ (x) − fm−1,θ (x)| ≤ diam Ik+1,Rω (θ) x∈Ii,θ ≤ diam (R((k + 1)∗ )) ≤ 2−|k+1| ≤ 2 · 2−|k| = 2 · 2−|b (θ,m−1)| . This ends the proof of the proposition in this subcase. Subcase 3.2. k ≥ 0 and θ ∈ Bαk (k ∗ ) \Bαk+1 (k ∗ ) . We start by computing fm−1,θ (Ii,θ ) . By Definition 7.2 and the definition of the maps gk,θ for k ≥ 0 (Definition 4.2), fm−1,θ (Ii,θ ) ⊂ fm−1,θ (Ik,θ ) = gk,θ (Ik,θ ) = {γk+1 (Rω (θ))}. Analogously, if i ≥ 0 and θ ∈ Bαi (i∗ ) \Bαi+1 (i∗ ) , fm,θ (Ii,θ ) = gi,θ (Ii,θ ) = {γi+1 (Rω (θ))}. Then, by Lemma 10.1, kfm,θ − fm−1,θ k = fm,θ I − fm−1,θ I i,θ i,θ = γi+1 (Rω (θ)) − γk+1 (Rω (θ)) ≤ 2−|b (θ,m−1)| . Assume now that i < 0 or i ≥ 0 and θ ∈ Bαi+1 (i∗ ) . By (26), Definition 7.2 and Lemmas 4.3(b) and 4.5(b) Rω (θ) ∈ Bα|i+1| ((i + 1)∗ ) , and fm,θ (Ii,θ ) = gi,θ (Ii,θ ) ⊂ Ii+1,Rω (θ) . Moreover, if k + 1 < |i + 1|, by Lemmas 5.4(a) and 3.6(c), we have   fm−1,θ (Ii,θ ) = γk+1 (Rω (θ)) = γ|i+1|−1 (Rω (θ)) ⊂ Ii+1,Rω (θ) . Therefore, by Lemma 3.6(f), kfm,θ − fm−1,θ k = sup |fm,θ (x) − fm−1,θ (x)| x∈Ii,θ = sup fm,θ (x) − γ|i+1|−1 (Rω (θ)) x∈Ii,θ  ≤ diam Ii+1,Rω (θ) ≤ diam (R((i + 1)∗ )) ≤ 2−|i+1| < 2−(k+1) < 2−|b (θ,m−1)| . So, to end the proof of the proposition we have to show that, in this subcase, k+1 < |i + 1|. To prove this, notice that when i ≥ 0, k+1 = |k|+1 < |i|+1 = |i + 1|. 62 ` F. MANOSAS, LL. ALSEDA, ˜ AND L. MORALES So, assume by way of contradiction that i < 0 and k + 1 = |i + 1| (recall that k + 1 ≤ |i + 1|). Then, k + 1 = −(i + 1) and, hence, Rω (θ) ∈ Rω (Bαk (k ∗ )) = Bαk ((k + 1)∗ ) , and Rω (θ) ∈ Bα|i+1| ((i + 1)∗ ) = Bαk+1 ((−(k + 1))∗ ) ⊂ B−(k+1) ((−(k + 1))∗ ) , which is a contradiction by Definition 3.4(R.2).  References [1] Roberta Fabbri, Tobias J¨ ager, Russell Johnson, and Gerhard Keller. A Sharkovskii-type theorem for minimally forced interval maps. Topol. Methods Nonlinear Anal., 26(1):163–188, 2005. ˇ [2] O. M. Sarkovs 0 ki˘ ı. Co-existence of cycles of a continuous mapping of the line into itself. ˘ 16:61–71, 1964. Ukrain. Mat. Z., [3] A. N. Sharkovski˘ı. Coexistence of cycles of a continuous map of the line into itself. In Thirty years after Sharkovski˘ı’s theorem: new perspectives (Murcia, 1994), volume 8 of World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., pages 1–11. World Sci. Publ., River Edge, NJ, 1995. Translated by J. Tolosa, Reprint of the paper reviewed in MR1361914 (96j:58058). Departament de Matema ` tiques and Centre de Recerca Matema` tica Edifici Cc, Uni- versitat Auto` noma de Barcelona, 08913 Cerdanyola del Valle`s, Barcelona, Spain E-mail address: alseda@mat.uab.cat Departament de Matema ` tiques, Edifici Cc, Universitat Auto ` noma de Barcelona, 08913 Cerdanyola del Valle `s, Barcelona, Spain E-mail address: manyosas@mat.uab.cat Departament de Matema ` tiques, Edifici Cc, Universitat Auto ` noma de Barcelona, 08913 Cerdanyola del Valle `s, Barcelona, Spain E-mail address: mleo@mat.uab.cat