Robustly chain transitive diffeomorphisms
- Manseob Lee^{1}Email author
https://doi.org/10.1186/s13660-015-0752-y
© Lee 2015
Received: 29 December 2014
Accepted: 9 July 2015
Published: 23 July 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.
Keywords
MSC
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 Ω-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 [1–4]). 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 Λ be a closed f-invariant set.
In [6], Mañé proved that if a diffeomorphism on two-dimensional \(C^{\infty}\) manifolds is robustly transitive, then it is hyperbolic, and Díaz et al. [7] proved that if a diffeomorphism on three-dimensional \(C^{\infty}\) 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 δ-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 Λ 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 Example 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 Λ is robustly chain transitive if there are a \(C^{1}\)-neighborhood \(\mathcal{U}(f)\) and a neighborhood U of Λ 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
- (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 Theorem 7.4 in [10]). To prove Theorem 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 Lemma 6 in [11].
Lemma 2.1
If \(f|_{\Lambda}\) is chain transitive, then \(f|_{\Lambda}\) has neither sinks nor sources.
Proof
Theorem 2.2
(Corollary 2.19 in [12])
- (1)
either f admits an l-dominated splitting along the orbit of x;
- (2)
or, for any neighborhood U of the orbit of x, there exists an ϵ-perturbation 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.
Lemma 2.3
(Theorem 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 Λ is the Hausdorff limit of a sequence of periodic points if and only if Λ is chain transitive.
Lemma 2.4
If \(f|_{\Lambda}\) is robustly chain transitive, then Λ admits a dominated splitting.
Proof
By Mañé (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
Let us recall Mañé’s ergodic closing lemma in [6]. For any \(\epsilon>0\), let \(B_{\epsilon}(f, x)\) an ϵ-tubular neighborhood of f-orbit 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 f-invariant probability measure μ, we have \(\mu(\Sigma_{f})=1\).
Proof of Theorem 1.1
Declarations
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. 2014R1A1A1A05002124).
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