axioms Article Generic Homeomorphisms with Shadowing of One-Dimensional Continua Alfonso Artigue 1, * and Gonzalo Cousillas 2, * 1 Departamento de Matemática y Estadística del Litoral, Universidad de la República, Salto 50000, Uruguay 2 Instituto de Matemática y Estadística “Rafael Laguardia”, Facultad de Ingeniería, Universidad de la República, Montevideo 11200, Uruguay * Correspondence: artigue@unorte.edu.uy (A.A.); gcousillas@fing.edu.uy (G.C.) Received: 7 May 2019; Accepted: 24 May 2019; Published: 26 May 2019 Abstract: In this article, we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property. Keywords: shadowing property; generic dynamics; one-dimensional dynamics MSC: 37E05; 37C50; 37C20 1. Introduction Let ( X, dist) be a compact metric space and denote by H( X ) the space of homeomorphisms f : X → X with the C0 distance distC0 ( 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 ([1], Theorem 1) or a Cantor set ([2], 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 [3,4], 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 (invariant under conjugacy, as the shadowing property) is reduced to determine whether a representative of this class has the property or not. In Theorem 2, 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 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 2 depends on Propositions 2 and 3, 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: if a homeomorphism has a Gδ dense conjugacy class, does it have the shadowing property? Axioms 2019, 8, 66; doi:10.3390/axioms8020066 www.mdpi.com/journal/axioms Axioms 2019, 8, 66 2 of 6 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 show the following result. Theorem 1. There is f ∗ ∈ H+ ( I ) whose conjugacy class is a Gδ dense subset of H + ( I ). Remark 1. The generic dynamics of circle homeomorphisms is studied in detail in [5], Theorem 9.1. The proof of Theorem 1 follows the same ideas. As we could not find this result in the literature, we include the details. To prove Theorem 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 a wandering interval. Following [6], 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 f r : [ a, b] → [ a, b] such that ( a, b) is an [ a,b] r-interval. Analogously, we consider f l with ( a, b) an l-interval. n For n ≥ 0 and 0 ≤ k < 3 , define the closed interval h i J (n, k ) = 3k +1 , 3k+2 . 3n +1 3n +1 For x in the ternary Cantor set, define f ∗ ( x ) = x. In another case, there is a minimum integer n x ≥ 0 such that x ∈ J (n x , k ) for some 0 ≤ k < 3nx and define ( J (n x ,k ) fl (x) if n x is odd, f ∗ (x) = J (n x ,k ) fr (x) if n x 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. From [7], Theorem 8, we know that f ∗ , and every homeomorphism conjugate to f ∗ , has the shadowing property. 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 1. 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 ∗ . Axioms 2019, 8, 66 3 of 6 To prove the converse, suppose that g satisfies Pε for all ε > 0. From 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 the 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 . In addition, 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 h j/k give rise to a conjugacy h of g and f ∗ in the whole interval [0, 1] and the proof ends. Proof of Theorem 1. Given n ≥ 1, let Un be the set of increasing homeomorphisms of I satisfying P1/n . Notice that Pε implies Pε0 for all ε0 > ε > 0. Thus, from Proposition 1 we have that the conjugacy class of f ∗ is the countable intersection n≥1 Un . To finish the proof, applying Baire’s Theorem, we show T 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 distC0 ( f m , gm ) < δ for all g ∈ V and gm ( xi ) > xi , gm (yi ) < yi for all i. This implies that ( xi , gm ( xi )) is contained in an r-interval for g and ( gm (yi ), yi ) is contained in an l-interval for g. For all g ∈ V and i odd, we have | gm ( xi ) − gm (yi+1 ))| ≤ | gm ( xi ) − f m ( xi )| + | f m ( xi ) − f m (yi+1 )| +| f m (yi+1 ) − gm (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. By finitely performing many 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 Axioms 2019, 8, 66 4 of 6 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 2. 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 1 and define Sn = Rn × Hn . Thus, ∩n≥1 ϕ− 1 n (Sn ) is a Gδ dense conjugacy class in H( X ). Remark 3. Notice that a representative g∗ of the Gδ dense conjugacy of Proposition 2 is obtained by considering a conjugate of f ∗ on each arc an of X. Now, we prove a sufficient condition for a homeomorphism to have the shadowing 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. Lemma 1. 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. Therefore, the proof ends. Proposition 3. 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 A x ⊂ X be a quasi-attractor with shadowing containing x. Let δx > 0 be given by Lemma 1 such that for every δx -pseudo-orbit { xn }n≥0 with x0 ∈ Bδx ( A x ) there is a point in A x that ε-shadows { xn }n≥0 . As X is compact, there is a finite sequence Axioms 2019, 8, 66 5 of 6 x1 , . . . , xk ∈ X such that ∪ik=1 Bδi ( Ai ) = X, where Ai = A xi 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 ( A j ). Then, there is a point in A j that ε-shadows { xn }n≥0 and the proof ends. Let Y ⊂ R2 be the union of • the circle arc x2 + y2 = 1, y ≤ 0; • the horizontal segment [−1, 1] × {0}; and • the vertical segments {−1 + n2 } × [0, 1/n], for n ≥ 1. See Figure 2. Figure 2. The continuum Y can be decomposed as a union of arcs as in Proposition 2. Theorem 2. 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 shadowing property is generic in H(Y ). Proof. The continuum Y satisfies the hypothesis of Proposition 2. Indeed, the conditions (1) and (2) are directly 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 2 holds. Therefore, H(Y ) contains a Gδ dense conjugacy class. As explained in Remark 3, 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, we know that g∗ : an → an has the shadowing property. By construction, each an is a quasi-attractor for g∗ . Since the arcs an cover Y, we can apply Proposition 3 to conclude that g∗ has the shadowing property. Author Contributions: Both authors contributed equally to this work. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. References 1. Pilyugin, S.Y.; Plamenevskaya, O.B. Shadowing is generic. Topol. Appl. 1999, 97, 253–266. [CrossRef] 2. Bernardes, N.C., Jr.; Darji, U.B. 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