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Robustly chain transitive diffeomorphisms
Journal of Inequalities and Applications volumeÂ 2015, ArticleÂ number:Â 230 (2015)
Abstract
In this paper, we discuss the robustly chain transitive set, and show that the robustly chain transitive set is hyperbolic if and only if every periodic points in the set is hyperbolic and has the same index.
1 Introduction
In the theory of dynamical systems one has been to describe and characterize systems exhibiting dynamical properties that are preserved under small perturbations. It is related to the stability theory. In fact, structurally stable systems and Î©stable systems have been the main objects of interests in the global qualitative theory of dynamical systems and they are characterized as the hyperbolic ones (see [1â€“4]). Thus, in differentiable dynamical systems, the robustness property is a very interesting topic. Let us consider more details. Let M be a closed \(C^{\infty}\) Riemannian manifold, and let \(\operatorname{Diff}(M)\) be the space of diffeomorphisms of M endowed with the \(C^{1}\)topology. Denote by d the distance on M induced from a Riemannian metric \(\\cdot\\) on the tangent bundle TM. Let \(f\in\operatorname{Diff}(M)\) and Î› be a closed finvariant set.
The set Î› is transitive if there is a point \(x\in\Lambda\) such that \(\omega(x)=\Lambda\). Here \(\omega(x)\) is the forward limit set of x. We say that Î› is locally maximal if there is a neighborhood U of Î› such that
We say that the set Î› is robustly transitive if there are a \(C^{1}\)neighborhood \(\mathcal{U}(f)\) of f and a neighborhood U of Î› such that for any \(g\in\mathcal{U}(f)\), \(\Lambda_{g}(U)=\bigcap_{n\in\mathbb{Z}}g^{n}(U)\) is transitive. Here \(f_{\Lambda}\) is transitive means that f is transitive in Î›. If \(\Lambda=M\) then we say that f is robustly transitive. We say that Î› is hyperbolic if the tangent bundle \(T_{\Lambda}M\) has a Dfinvariant splitting \(E^{s}\oplus E^{u}\) and there exist constants \(C>0\) and \(0<\lambda<1\) such that
for all \(x\in\Lambda\) and \(n\geq0\). If \(\Lambda=M\) then we say that f is Anosov. Although f is robustly transitive, we can find that f is not Anosov. In fact, MaÃ±Ã© [5] showed that there exists a diffeomorphism f on the threedimensional torus \(\mathbb{T}^{3}\) that satisfies: there is a \(C^{1}\)neighborhood \(\mathcal{U}(f)\) of f such that every \(g\in\mathcal{U}(f)\) is transitive, but not Anosov.
In [6], MaÃ±Ã© proved that if a diffeomorphism on twodimensional \(C^{\infty}\) manifolds is robustly transitive, then it is hyperbolic, and DÃaz et al. [7] proved that if a diffeomorphism on threedimensional \(C^{\infty}\) manifolds is robustly transitive then it is partially hyperbolic. Also, in [8], the authors proved that for \(C^{\infty}\) manifolds of any dimension, if a diffeomorphism is robustly transitive, then it admits a dominated splitting.
From the facts, we study the relation between the robustly chain transitive and hyperbolicity. For given \(x, y\in M\), we write \(x\rightsquigarrow y\) if for any \(\delta>0\), there is a finite Î´pseudo orbit \(\{x_{i}\}_{i=0}^{n}\) (\(n\geq1\)) of f such that \(x_{0}=x\) and \(x_{n} =y\). For any \(x, y\in\Lambda\), we write \(x\rightsquigarrow_{\Lambda} y\) if \(x\rightsquigarrow y\) and \(\{x_{i}\}_{i=0}^{n}\subset\Lambda\) (\(n\geq1\)). We say that the set Î› is chain transitive (or, \(f_{\Lambda}\) is chain transitive) if for any \(x, y\in\Lambda\), \(x\rightsquigarrow_{\Lambda} y\). Note that by the definition, a transitive set is a chain transitive set, but the converse is not true (see ExampleÂ 1.5 in [9]). In this paper, we study robustly chain transitive sets for a diffeomorphism. It is weaker notion of the robustly transitivity. Let \(p\in P(f)\) be a hyperbolic point. Denote by \(\operatorname{index}(p)=\operatorname{dim}W^{s}(p)\). We say that the set Î› is robustly chain transitive if there are a \(C^{1}\)neighborhood \(\mathcal{U}(f)\) and a neighborhood U of Î› such that for any \(g\in\mathcal{U}(f)\), \(\Lambda_{g}(U)=\bigcap_{n\in\mathbb{Z}}g^{n}(U)\) is chain transitive. Then we have the following.
Theorem 1.1
Let \(f_{\Lambda}\) be robustly chain transitive in U. Then following conditions are equivalent:

(a)
there is a \(C^{1}\)neighborhood \(\mathcal{U}(f)\) of f such that for any \(g\in\mathcal{U}(f)\), any periodic point of \(\Lambda_{g}(U)\) is hyperbolic and has the same index;

(b)
there is a \(C^{1}\)neighborhood \(\mathcal{U}(f)\) of f such that for any \(g\in\mathcal{U}(f)\), \(\Lambda_{g}(U)\) is hyperbolic.
2 Proof of Theorem 1.1
It is clear that (a) follows from (b) by the local stability of hyperbolic basic set (see TheoremÂ 7.4 in [10]). To prove TheoremÂ 1.1, we show from (a) to (b). We say that \(p\in P(f)\) with period \(\pi(p)\) is a sink if all the eigenvalues of \(D_{p}f^{\pi(p)}\) are less than 1, and \(p\in P(f)\) with period \(\pi(p)\) is a source if all eigenvalues of \(D_{p}f^{\pi(p)}\) is greater than 1. The following is the version for diffeomorphisms of the result by LemmaÂ 6 in [11].
Lemma 2.1
If \(f_{\Lambda}\) is chain transitive, then \(f_{\Lambda}\) has neither sinks nor sources.
Proof
Let p be a sink. Then there exist \(\epsilon>0\) and \(\lambda<1\) such that if \(d(x, p)<\epsilon\) then \(d(f^{i}(x), p)<\lambda d(x, p)\) for all \(i\geq1\). Take \(y\in\Lambda\) such that \(d(y, p)\geq2\epsilon\). For any \(\delta>0\), let \(\xi=\{p=x_{0}, x_{1}, \ldots, x_{m} =y\}\) (\(m\geq1\)) be a Î´pseudo orbit of f such that \(x_{i}\in\Lambda\). For simplicity, we may assume that \(f(p)=p\). Then we have \(d(p, x_{1})<\delta\), and \(d(p, x_{2})\leq d(p, f(x_{1}))+d(f(x_{1}), x_{2})< \lambda d(p, x_{1})+\delta<\delta(\lambda+1)\). Thus we obtain
Put \(\eta=\delta/(1\lambda)\). Then if Î´ is sufficiently small then we can make \(\eta<\epsilon\). This is a contradiction since \(d(y, p)\geq2\epsilon\).â€ƒâ–¡
Theorem 2.2
(CorollaryÂ 2.19 in [12])
For any \(\epsilon>0\), there are two integers l and n such that for any periodic point x of period \(p(x)\geq n\):

(1)
either f admits an ldominated splitting along the orbit of x;

(2)
or, for any neighborhood U of the orbit of x, there exists an Ïµperturbation g of f in the \(C^{1}\)topology, coinciding with f outside U and on the orbit of x, and such that x is a source or a sink of g for which the differential \(D_{x}g^{p(x)}\) has all eigenvalues real with the same modulus.
We say that the Hausdorff distance between two closed subsets A and B of M is given by
Lemma 2.3
(TheoremÂ 4 in [9])
There is a residual set \(\mathcal{G}\subset\operatorname{Diff}(M)\) such that for any \(f\in\mathcal{G}\), a compact invariant set Î› is the Hausdorff limit of a sequence of periodic points if and only if Î› is chain transitive.
Lemma 2.4
If \(f_{\Lambda}\) is robustly chain transitive, then Î› admits a dominated splitting.
Proof
Let \(f_{\Lambda}\) be robustly chain transitive. Then there are a \(C^{1}\)neighborhood \(\mathcal{U}(f)\) of f and a neighborhood U of Î› such that for any \(g\in\mathcal{U}(f)\), \(g_{\Lambda_{g}(U)}\) is chain transitive. By LemmaÂ 2.3, take \(\mathcal{U}_{0}(f)=\mathcal{U}(f)\cap \mathcal{G}\). Then there exist \(f_{n}\in\mathcal{U}_{0}(f)\) with \(f_{n} \to f\) and \(\operatorname{Orb}_{f_{n}}(p_{n})\) of \(f_{n}\) such that
Since \(f_{\Lambda}\) is robustly chain transitive, by LemmaÂ 2.1, \(f_{\Lambda}\) does not contains neither sinks nor sources. By TheoremÂ 2.2, \(f_{n}\) admits an ldominated splitting over \(\operatorname{Orb}_{f_{n}}(p_{n})\) with l independent n. Thus Î› admits an ldominated splitting.â€ƒâ–¡
By MaÃ±Ã© (see [6]), the family of periodic sequences of linear isomorphisms of \(\mathbb{R}^{\operatorname{dim}M}\) generated by Dg (g close to f) along the hyperbolic periodic point \(q\in\Lambda_{g}(U)\cap P(g)\) is uniformly hyperbolic. This means that there is \(\epsilon>0\) such that for any g \(C^{1}\)nearby \(f,q\in\Lambda_{g}(U)\cap P(g)\) and any sequence of linear maps \(A_{i}:T_{g^{i}(q)}M\to T_{g^{i+1}(q)}M\) with \(\A_{i}D_{g^{i}(q)}g\<\epsilon\) (\(i=1, 2,\ldots, \pi(q)\)), \(\prod_{i=0}^{\pi(q)1}A_{i}\) is hyperbolic. By Proposition II.1 in [6], we have the following.
Lemma 2.5
Suppose that there is a \(C^{1}\)neighborhood \(\mathcal{U}(f)\) of f such that for any \(g\in\mathcal{U}(f)\), any periodic point of \(\Lambda_{g}(U)\) is hyperbolic. Then there are constants \(C>0\), \(0<\lambda<1\) and \(m>0\) such that if for any \(p\in\Lambda_{g}(U)\cap P(g)\) has minimum period \(\pi(p)\geq m\) then
where \(k=[\pi(p)/m]\).
Let us recall MaÃ±Ã©â€™s ergodic closing lemma in [6]. For any \(\epsilon>0\), let \(B_{\epsilon}(f, x)\) an Ïµtubular neighborhood of forbit of x, i.e., \(B_{\epsilon}(f, x)=\{y\in M: d(f^{n} (x), y)<\epsilon \mbox{ for some } n\in\mathbb{Z}\}\). Let \(\Sigma_{f}\) be the set of points \(x\in M\) such that for any \(C^{1}\)neighborhood \(\mathcal{U}(f)\) of f and \(\epsilon>0\), there are \(g\in\mathcal{U}(f)\) and \(y\in P(g)\) satisfying \(g=f\) on \(M\setminus B_{\epsilon}(f, x)\) and \(d(f^{i}(x), g^{i}(y))\leq \epsilon\) for \(0\leq i\leq\pi(y)\).
Remark 2.6
(Theorem A in [6])
For any finvariant probability measure Î¼, we have \(\mu(\Sigma_{f})=1\).
Proof of TheoremÂ 1.1
By assumption, \(f_{\Lambda}\) is robustly chain transitive. Then by LemmaÂ 2.4, Î› admits a dominated splitting \(E\oplus F\). We will finish the proof of TheoremÂ 1.1, we show that
for all \(x\in \Lambda\). Then the splitting is hyperbolic. To prove, we consider \(\liminf_{n \to\infty} \ D_{x}f^{n}_{E} \ = 0\) (other case is similar). It is enough to show that for any \(x\in \Lambda\), there exists \(n=n(x)>0\) such that
We will derive a contraction. If it is not true, then there is \(x\in\Lambda\) such that
for all \(n\geq0\). Thus
for all \(n\geq0\). Define a probability measure
Then there exists \(\mu_{n_{k}}\) (\(k\geq0\)) such that \(\mu_{n_{k}}\rightarrow \mu_{0}\in\mathcal{M}_{f}(M)\), as \(k\rightarrow\infty\), where M is compact metric space. Thus
By MaÃ±Ã© ([6], p.521),
where \(\mu_{0}\) is a finvariant measure. Let
and \(\Sigma_{f}\) = {\(x\in M : d(f^{n} (x), y)<\epsilon\), there exist \(g\in \mathcal{U}(f)\) and \(y\in P(g)\) such that \(g=f\) on \(M\setminus B_{\epsilon}(f, x)\) and \(d(f^{i}(x), f^{i}(y))\leq\epsilon\) for \(0\leq i\leq \pi(y)\)}.
For any \(\mu\in\mathcal{M}_{f}(M)\), \(\mu(\Sigma_{f})=1\). Then, for any \(\mu\in\mathcal{M}_{f}(\Lambda)\),
since \(\mu(\Lambda)=1\) and \(\mu(\Sigma_{f})=1\). Thus, \(\Lambda =\Lambda\cap\Sigma(f)\) almost everywhere. Therefore,
By Birkhoffâ€™s theorem, and the ergodic closing lemma, we can take \(z_{0}\in\Lambda\cap\Sigma(f)\) such that
By LemmaÂ 2.5, this is a contradiction. Thus by LemmaÂ 2.5, \(z_{0}\notin P(f)\).
Let \(C>0\), \(m>0\), and \(\lambda\in(0, 1)\) be given by LemmaÂ 2.5 and take \(\lambda<\lambda_{0}<1\) and \(n_{0}>0\) such that
Then, by MaÃ±Ã©â€™s ergodic closing lemma, we can find \(g \in \mathcal{V}_{0}(f)\), \(g = f\) on \(M \setminus U_{j}\) and \(r_{g} \in \Lambda_{g} \cap P(g)\) near by \(z_{0}\). By assumption, for any \(p\in\Lambda_{g}(U)\cap P(g)\) we know that
since \(g = f\) on \(M \setminus U_{j}\). Since \(f_{\Lambda}\) is robustly chain transitive, we can construct \(h \in\mathcal{ V}_{0}(f)\) (\(\subset\mathcal{V}(f)\)) \(C^{1}\)nearby g such that
(see [6], pp.523524). On the other hand, by LemmaÂ 2.5, we see that
We can choose the period \(\pi(r_{h})\) (\(> n_{0}\)) of \(r_{h}\) as large as \(\lambda_{0}^{k} \geq C \lambda^{k}\). Here \(k = [\pi(r_{h})/m]\). This is a contradiction. Thus,
for all \(x \in\Lambda\). Therefore, Î› is hyperbolic. This completes the proof of TheoremÂ 1.1.â€ƒâ–¡
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Acknowledgements
ML is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No.Â 2014R1A1A1A05002124).
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Lee, M. Robustly chain transitive diffeomorphisms. J Inequal Appl 2015, 230 (2015). https://doi.org/10.1186/s136600150752y
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DOI: https://doi.org/10.1186/s136600150752y