This is a preprint of: “Generic homeomorphisms with shadowing of one-dimensional continua”, Alfonso Artigue, Gonzalo Cousillas, Axioms, vol. 8, 66:1–6, 2019. DOI: [10.3390/axioms8020066] GENERIC HOMEOMORPHISMS WITH SHADOWING OF ONE-DIMENSIONAL CONTINUA ALFONSO ARTIGUE AND GONZALO COUSILLAS Abstract. In this article we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property. 1. Introduction Let (X, dist) be a compact metric space and denote by H(X) the space of home- omorphisms f : X → X with the C 0 distance distC 0 (f, g) = sup{dist(f (x), g(x)), dist(f −1 (x), g −1 (x)) : x ∈ X}. A property is said to be generic if it holds on a residual subset of H(X). Recall that a set is Gδ if it is a countable intersection of open sets and it is residual if it contains a dense Gδ subset. For instance, it is known that the shadowing property is generic for X a compact manifold [6, Theorem 1] or a Cantor set [3, Theorem 4.3]. Recall that f ∈ H(X) has the shadowing property if for all ε > 0 there is δ > 0 such that if {xi }i∈Z is a δ-pseudo orbit then there is y ∈ X such that dist(f i (y), xi ) < ε for all i ∈ Z. We say that {xi }i∈Z is a δ-pseudo orbit if dist(f (xi ), xi+1 ) < δ for all i ∈ Z. A remarkable result, proved in [1, 4], states that if X is a Cantor set then there is a homeomorphism of X whose conjugacy class is a dense Gδ subset of H(X). That is, a generic homeomorphism of a Cantor set is conjugate to this special homeomorphism. We say that f, g ∈ H(X) are conjugate if there is h ∈ H(X) such that f ◦ h = h ◦ g and the conjugacy class of f is the set of all the homeomorphisms conjugate to f . This result gives rise to a natural question: besides the Cantor set, which compact metric spaces have a Gδ dense conjugacy class? On a space with a Gδ dense conjugacy class, the study of generic properties (invari- ant under conjugacy, as the shadowing property) is reduced to determine whether a representative of this class has the property or not. In Theorem 3.5 we show that there are one-dimensional plane continua with a Gδ dense conjugacy class whose members have the shadowing property. The proof of this result is based on Theorem 2.1 where we show that for a compact interval I there is a Gδ conjugacy class in H(I) which is dense in the subset of orientation preserving homeomorphisms of I. In addition, the proof of Theorem 3.5 depends on Propositions 3.1 and 3.4 where we give sufficient conditions for the existence of a residual conjugacy class and for a homeomorphism to have the shadowing property, respectively. The following open question has an affirmative answer in the examples known by the authors: 1 2 A. ARTIGUE AND G. COUSILLAS if a homeomorphism has a Gδ dense conjugacy class, does it have the shadowing property? 2. Generic dynamics on a closed segment Let I = [0, 1] and define H+ (I) = {f ∈ H(I) : f preserves orientation}. In this section we will show the next result. Theorem 2.1. There is f∗ ∈ H+ (I) whose conjugacy class is a Gδ dense subset of H+ (I). Remark 2.2. The generic dynamics of circle homeomorphisms is studied in detail in [2, Theorem 9.1]. The proof of Theorem 2.1 follows the same ideas. As we could not find this result in the literature we include the details. To prove Theorem 2.1 we start by defining the homeomorphism f∗ . For this purpose we introduce some definitions. For f ∈ H+ (I) let fix(f ) = {x ∈ X : f (x) = x}. A connected component of I \fix(f ) will be called wandering interval. Following [7], we say that a wandering interval (a, b) is an r-interval if limn→+∞ f n (x) = b for all x ∈ (a, b). Analogously, it is an l-interval if limn→+∞ f n (x) = a for all [a,b] x ∈ (a, b). For each interval [a, b] fix a homeomorphism fr : [a, b] → [a, b] such [a,b] that (a, b) is an r-interval. Analogously, we consider fl with (a, b) an l-interval. For n ≥ 0 and 0 ≤ k < 3n define the closed interval   J(n, k) = 3k+1 3k+2 3n+1 , 3n+1 . For x in the ternary Cantor set define f∗ (x) = x. In other case, there is a minimum integer nx ≥ 0 such that x ∈ J(nx , k) for some 0 ≤ k < 3nx and define ( J(n ,k) fl x (x) if nx is odd, f∗ (x) = J(n ,k) fr x (x) if nx is even. For example, ( 31 , 23 ) is an r-interval, while ( 312 , 322 ) and ( 372 , 382 ) are l-intervals. See Figure 1. Figure 1. A sketch of the phase diagram of f∗ . Remark 2.3. From [5, Theorem 8] we know that f∗ , and every homeomorphism conjugate to f∗ , has the shadowing property. GENERIC HOMEOMORPHISMS WITH SHADOWING OF ONE-DIMENSIONAL CONTINUA 3 The next result gives a useful characterization of the conjugacy class of f∗ . Given ε > 0 we say that g ∈ H+ (I) satisfies the property Pε if there are intervals Ji = (ai , bi ), i = 1, . . . , n, such that: (1) 0 < a1 < b1 < a2 < b2 < a3 < · · · < bn < 1, (2) Ji is an r-interval for i odd and an l-interval for i even, (3) max{a1 , 1 − bn } < ε and max{ai+1 − bi : 1 ≤ i < n} < ε. Proposition 2.4. A homeomorphism g ∈ H+ (I) is conjugate to f∗ if and only if it satisfies Pε for all ε > 0. Proof. The direct part of the proof is clear from the construction of f∗ . To prove the converse suppose that g satisfies Pε for all ε > 0. From the condition (3) we see that fix(g) is totally disconnected. Suppose that p ∈ I is an isolated fixed point. If p = 0 then there is a wandering interval (0, x). Taking ε ∈ (0, x) we have a contradiction with (3), because a1 < ε. Analogously we show that p cannot be 1. If p ∈ (0, 1) then p is in the boundary of two wandering intervals. Taking ε smaller than the length of these intervals we contradict (1) and (3). Thus, fix(g) has no isolated point and is a Cantor set. Condition (2) (applied for a suitable ε small) implies that between two wandering intervals there is an r-interval and an l-interval. Let R and L be the families of r-intervals and l-intervals of g respectively. We define an order in R ∪ L in the following way: Iα < Iβ if x < y for all x ∈ Iα , y ∈ Iβ . We will make the conjugacy by induction. For the first step name I1/2 ∈ R which satisfies diam(I1/2 ) ≥ diam(I) for every I ∈ R. In case that there exists more than one interval which verifies this condition we choose any of them. Let Jc be a wandering interval of f∗ such that c is the midpoint of Jc . By construction, J1/2 is an r-interval of f∗ , thus we can consider a conjugacy h1/2 : I1/2 → J1/2 of g and f∗ restricted to these intervals. Notice that 1/6 and 5/6 are the midpoints of (1/9, 2/9) and (7/9, 8/9) respectively. Take I1/6 ∈ L satisfying I1/6 < I1/2 and diam(I1/6 ) ≥ diam(I) for every I ∈ L such that I < I1/2 . Also, take I5/6 ∈ L satisfying I1/2 < I5/6 and diam(I5/6 ) ≥ diam(I) for every I ∈ L such that I > I1/2 . Then consider h1/6 : I1/6 → J1/6 to be a conjugacy from g to f∗ restricted to the corresponding intervals. Similarly, define h5/6 . Then we go on defining 2k−1 homeomorphisms on each step. If k − 1 is even, we choose r-intervals, otherwise we choose l-intervals. Notice that since in each step we choose the largest interval of the r or l-intervals of g, every wandering interval of g is eventually chosen. In this way, the conjugacies hj/k give rise to a conjugacy h of g and f∗ in the whole interval [0, 1] and the proof ends.  Proof of Theorem 2.1. Given n ≥ 1, let Un be the set of increasing homeomor- phisms of I satisfying P1/n . Notice that Pε implies Pε0 for all ε0 > ε > 0. Thus, from Proposition T 2.4 we have that the conjugacy class of f∗ is the countable inter- section n≥1 Un . To finish the proof, applying Baire’s Theorem, we will show that each Un is open and dense in H+ (I). To prove that Un is open consider f ∈ Un . It is clear that there is δ > 0 such that f ∈ Un−4δ . Consider the intervals (ai , bi ) from the definition of property Pε , for ε = 1/n. For each odd i = 1, . . . , n, take xi ∈ (ai , ai + δ) and for i even take yi ∈ (bi − δ, bi ). Consider m ∈ N large such that f m (xi ) ∈ (bi − δ, bi ) and f m (yi ) ∈ (ai , ai + δ) for all i. Take a neighborhood V of f such that distC 0 (f m , g m ) < δ for all g ∈ V and g m (xi ) > xi , g m (yi ) < yi for all i. This implies that (xi , g m (xi )) is 4 A. ARTIGUE AND G. COUSILLAS contained in an r-interval for g and (g m (yi ), yi ) is contained in an l-interval for g. For all g ∈ V and i odd we have |g m (xi ) − g m (yi+1 ))| ≤ |g m (xi ) − f m (xi )| + |f m (xi ) − f m (yi+1 )| +|f m (yi+1 ) − g m (yi+1 )| < δ + |f m (xi ) − bi | + |bi − ai+1 | + |ai+1 − f m (yi+1 )| + δ < 2δ + (1/n − 4δ) + 2δ = 1/n. Arguing analogously for i even, we conclude that g ∈ Un and Un is open. To prove that Un is dense in H+ (I) the following remark is sufficient. Given f ∈ H+ (I), p ∈ fix(f ) ∩ (0, 1) and δ > 0 small, we can define g ∈ H+ (I) close to f such that: • f |[0,p] and g|[0,p−δ] are conjugate, • f |[p,1] and g|[p+δ,1] are conjugate and • g has an r or l-interval at [p − δ, p + δ]. That is, a fixed point can be exploded into a small wandering interval with an arbitrarily small perturbation. Performing finitely many of such explosions the density of Un is obtained.  3. Genericity on a plane one-dimensional continuum In this section we show that there are some particular one-dimensional plane continua with a Gδ dense conjugacy class whose members have the shadowing property. We start with a sufficient condition for the existence of a Gδ dense conjugacy class. An open subset U ⊂ X is a free arc if it is homeomorphic to R. Proposition 3.1. If X is a compact metric space such that (1) X = ∪n≥1 an , where each an is a compact arc with extreme points pn , qn ∈ X for all n ≥ 1, (2) an \ {pn , qn } is a free arc for all n ≥ 1 and (3) for all f ∈ H(X) it holds that f (an ) = an and pn , qn ∈ fix(f ) for all n ≥ 1 then H(X) has a Gδ dense conjugacy class. Proof. For each n ≥ 1 let Xn = clos(X \ an ) and define Hn = {f ∈ H(Xn ) : pn , qn ∈ fix(f )} and the map ϕn : H(X) → H+ (an ) × Hn as ϕn (f ) = f |an × f |Xn . In H+ (an ) × Hn we consider the product topology. It is clear that ϕn is a homeomorphism for each n ≥ 1. Let Rn be the Gδ dense conjugacy class of H+ (an ) given by Theorem 2.1 and define Sn = Rn × Hn . Thus, ∩n≥1 ϕ−1 n (Sn ) is a Gδ dense conjugacy class in H(X).  Remark 3.2. Notice that a representative g∗ of the Gδ dense conjugacy of Propo- sition 3.1 is obtained by considering a conjugate of f∗ on each arc an of X. Now we will prove a sufficient condition for a homeomorphism to have the shad- owing property. For this purpose we need some definitions and a lemma. Suppose that (X, dist) is a compact metric space and take f ∈ H(X). A compact f -invariant subset A ⊂ X is a quasi-attractor if for every open neighborhood U of A there is an open subset V ⊂ U such that A ⊂ V and clos(f (V )) ⊂ V . If in addition f : A → A has the shadowing property we say that A is a quasi-attractor with shadowing. GENERIC HOMEOMORPHISMS WITH SHADOWING OF ONE-DIMENSIONAL CONTINUA 5 Lemma 3.3. If A ⊂ X is a quasi-attractor with shadowing then for all ε > 0 there is δ > 0 such that if {xn }n≥0 is a δ-pseudo-orbit with x0 ∈ Bδ (A) then there is y ∈ A that ε-shadows {xn }n≥0 . Proof. Given ε > 0 take δ1 > 0 such that every δ1 -pseudo-orbit in A is ε/2-shadowed by a point in A. Consider 0 < α < min{ε/2, δ1 /3} such that dist(a, b) < α implies dist(f (a), f (b)) < δ1 /3. Since A is a quasi-attractor, for U = Bα (A) there exists an open set V such that A ⊂ V ⊂ U and clos(f (V )) ⊂ V . Take δ ∈ (0, δ1 /3) such that Bδ (clos(f (V ))) ⊂ V . Suppose that {xn }n≥0 is a δ-pseudo-orbit with x0 ∈ Bδ (A). Since f (x0 ) ∈ f (V ) we have that x1 ∈ Bδ (f (V )) and x1 ∈ V . In this way we prove that xn ∈ V for all n ≥ 0. For each n ≥ 0 take yn ∈ A such that dist(yn , xn ) < α. We have that dist(f (yn ), yn+1 ) ≤ dist(f (yn ), f (xn )) + dist(f (xn ), xn+1 ) + dist(xn+1 , yn+1 ) ≤ δ1 /3 + δ + α < 3δ1 /3 = δ1 . This proves that {yn }n≥0 is a δ1 -pseudo-orbit contained in A. There exists z ∈ A that ε/2-shadows {yn }n≥0 . Thus dist(f n (z), xn ) ≤ dist(f n (z), yn ) + dist(yn , xn ) < ε/2 + α ≤ ε. And the proof ends.  Proposition 3.4. If every point of X belongs to a quasi-attractor with shadowing then f has shadowing. Proof. Suppose that ε > 0 is given. For each x ∈ X let Ax ⊂ X be a quasi-attractor with shadowing containing x. Let δx > 0 be given by Lemma 3.3 such that for every δx -pseudo-orbit {xn }n≥0 with x0 ∈ Bδx (Ax ) there is a point in Ax that ε-shadows {xn }n≥0 . As X is compact there is a finite sequence x1 , . . . , xk ∈ X such that ∪ki=1 Bδi (Ai ) = X where Ai = Axi and δi = δxi . If we take δ = min{δ1 , . . . , δk } we have that for every δ-pseudo-orbit {xn }n≥0 in X there is j such that x0 ∈ Bδj (Aj ). Then, there is a point in Aj that ε-shadows {xn }n≥0 and the proof ends.  Let Y ⊂ R2 be the union of: • the circle arc x2 + y 2 = 1, y ≤ 0, • the horizontal segment [−1, 1] × {0} and • the vertical segments {−1 + n2 } × [0, 1/n], for n ≥ 1. See Figure 2. Theorem 3.5. For the continuum Y there is a Gδ conjugacy class which is dense in H(Y ) and whose members have the shadowing property. In particular, the shad- owing property is generic in H(Y ). Proof. The continuum Y satisfies the hypothesis of Proposition 3.1. Indeed, the conditions (1) and (2) are direct from the construction of Y . Consider the points pn , p̃ indicated in Figure 2. It is clear that p̃ ∈ fix(f ) for all f ∈ H(Y ). This implies that a1 is invariant and p2 ∈ fix(f ). In turn, this implies that a2 is invariant under each f ∈ H(Y ). In this way it is shown that condition (3) of Proposition 3.1 holds. Therefore, H(Y ) contains a Gδ dense conjugacy class. As explained in Remark 3.2, a representative g∗ ∈ H(Y ) of this conjugacy class is obtained by taking a conjugate of f∗ on each arc an . It only remains to prove that g∗ has the shadowing property. By Remark 2.3 we know that g∗ : an → an has the shadowing property. By construction, each an is a quasi-attractor for g∗ . 6 A. ARTIGUE AND G. COUSILLAS Figure 2. The continuum Y can be decomposed as a union of arcs as in Proposition 3.1. Since the arcs an cover Y we can apply Proposition 3.4 to conclude that g∗ has the shadowing property.  References [1] Ethan Akin, Eli Glasner, and Benjamin Weiss. Generically there is but one self homeomor- phism of the Cantor set, In Trans. Amer. Math. Soc. 360 (2008), no. 7, 3613–3630. [2] Ethan Akin, Mike Hurley, and Judy A. Kennedy. 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E-mail address: artigue@unorte.edu.uy Instituto de Matemática y Estadística “Rafael Laguardia”, Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay. E-mail address: gcousillas@fing.edu.uy