The ergodic shadowing property from the robust and generic view point
 Manseob Lee^{1}Email author
https://doi.org/10.1186/168718472014170
© Lee; licensee Springer 2014
Received: 6 January 2014
Accepted: 29 May 2014
Published: 23 June 2014
Abstract
In this paper, we discuss that if a diffeomorphisms has the ${C}^{1}$stably ergodic shadowing property in a closed set, then it is a hyperbolic elementary set. Moreover, ${C}^{1}$generically: if a diffeomorphism has the ergodic shadowing property in a locally maximal closed set, then it is a hyperbolic basic set.
MSC:34D30, 37C20.
Keywords
ergodic shadowing shadowing locally maximal generic Anosov1 Introduction
Note that f has the ergodic shadowing property on Λ and f has the ergodic shadowing property in Λ are different notions. That is, the shadowing point is in M or Λ. In the first notion, the shadowing point is in M. In the second notion, the shadowing point is in Λ. In this paper we consider the latter case.
Now, we introduce the notion of the ${C}^{1}$stably ergodic shadowing property in a closed set.
 (i)
there is a neighborhood U of Λ and a ${C}^{1}$neighborhood $\mathcal{U}(f)$ of f such that ${\mathrm{\Lambda}}_{f}(U)=\mathrm{\Lambda}={\bigcap}_{n\in \mathbb{Z}}{f}^{n}(U)$ (that is, Λ is locally maximal);
 (ii)
for any $g\in \mathcal{U}(f)$, g has the ergodic shadowing property on ${\mathrm{\Lambda}}_{g}(U)={\bigcap}_{n\in \mathbb{Z}}{g}^{n}(U)$, where ${\mathrm{\Lambda}}_{g}(U)$ is the continuation of Λ.
for all $x\in \mathrm{\Lambda}$ and $n\ge 0$. If $\mathrm{\Lambda}=M$, then f is Anosov. We say that Λ is a basic set (resp. elementary set) if $f{}_{\mathrm{\Lambda}}$ is transitive (resp. mixing) and locally maximal. Note that if Λ is hyperbolic, then we can easily show that there is a periodic point such that the orbit of the periodic point is dense in the set. Then we get the following.
Theorem 1.2 [[5], Theorem 3.3]
Let Λ be a closed finvariant set. If f has the ${C}^{1}$stably ergodic shadowing property in Λ, then it is a hyperbolic elementary set.
Corollary 1.3 If f belongs to the ${C}^{1}$interior of the set of all diffeomorphisms having the ergodic shadowing property, then it is transitive Anosov.
We say that a subset $\mathcal{G}\subset Diff(M)$ is residual if contains the intersection of a countable family of open and dense subsets of $Diff(M)$; in this case is dense in $Diff(M)$. A property P is said to be ${C}^{1}$generic if P holds for all diffeomorphisms which belong to some residual subset of $Diff(M)$. We use the terminology ‘for ${C}^{1}$generic f’ to express ‘there is a residual subset $\mathcal{G}\subset Diff(M)$ such that, for any $f\in \mathcal{G}\dots $ .’ In [6], Abdenur and Díaz proved that if tame diffeomorphisms has the shadowing property, then it is hyperbolic. Still open is the question if ${C}^{1}$generically: f is shadowable, then is it hyperbolic?
Recently, Ahn et al. [7] have given a partial answer which is ${C}^{1}$generically: if a locally maximal homoclinic class is shadowing, then it is hyperbolic. Lee has shown in [8] that ${C}^{1}$generically: if f has the limit shadowing property on the homoclinic class, then it is hyperbolic. Inspired by this, we consider that ${C}^{1}$generically: f has the ergodic shadowing property in a locally maximal closed set. Then we have the following.
Theorem 1.4 For ${C}^{1}$generic f, if f has the ergodic shadowing property in a locally maximal closed set Λ, then it is a hyperbolic elementary set. Moreover, ${C}^{1}$generically: if f has the ergodic shadowing property, then it is transitive Anosov.
2 Proof of Theorem 1.4
Let $P(f)$ be the set of periodic points of f. If $f{}_{\mathrm{\Lambda}}$ is transitive, then every $p\in \mathrm{\Lambda}\cap P(f)$ is saddle, that is, there is no eigenvalues of ${D}_{p}{f}^{\pi (p)}$ with modulus equal to 1, at least one of them is greater than 1, at least one of them is smaller than 1, where $\pi (p)$ is the minimum period of p.
Lemma 2.1 [[2], Corollary 3.5]
If f has the ergodic shadowing property in Λ, then $f{}_{\mathrm{\Lambda}}$ is mixing.
By Lemma 2.1, f has the ergodic shadowing property in Λ, then $f{}_{\mathrm{\Lambda}}$ is mixing, and so $f{}_{\mathrm{\Lambda}}$ is transitive. Thus $p\in \mathrm{\Lambda}\cap P(f)$ is neither a sink nor a source.
Lemma 2.2 [[2], Lemma 3.2]
If f has the ergodic shadowing property in Λ, then f has a finite shadowing property in Λ.
We say that f has the finite shadowing property on Λ if for any $\u03f5>0$ there is $\delta >0$ such that, for any finite δpseudo orbit $\{{x}_{0},{x}_{1},\dots ,{x}_{n}\}\subset \mathrm{\Lambda}$, there is $y\in M$ such that $d({f}^{i}(y),{x}_{i})<\u03f5$ for all $0\le i<n$. In [[9], Lemma 1.1.1], Pilyugin showed that f has a finite shadowing shadowing property on Λ, then f has the shadowing property on Λ.
Lemma 2.3 Let f have the ergodic shadowing in Λ and Λ be locally maximal in U. Then the shadowing point taken from Λ.
Proof Let f have the ergodic shadowing property in Λ, and let U be a locally maximal of Λ. For any $\u03f5>0$, let $\delta >0$ be the number of the ergodic shadowing property of f. Take a sequence $\gamma ={\{{x}_{i}\}}_{i=0}^{n}$ ($n\ge 1$) such that γ is a δpseudo orbit of f and $\gamma \subset \mathrm{\Lambda}$. As in the proof of [[2], Lemma 3.1], there is a δpseudo orbit $\eta ={\{{x}_{i}\}}_{i=n}^{0}$ such that $\eta \subset \mathrm{\Lambda}$. Then we set $\xi =\{\dots ,\gamma ,\eta ,\gamma ,\eta ,\dots \}$ is a δergodic pseudo orbit of f. Clear that $\xi \subset \mathrm{\Lambda}$. Since f has the ergodic shadowing property in Λ, ξ can be ergodic shadowed by some point $y\in \mathrm{\Lambda}$. By Lemma 2.2, there is $\gamma \in \xi $ such that $d({f}^{i}(y),{x}_{i})<\u03f5$ for $0\le i\le n1$. By [[9], Lemma 1.1.1], f has the shadowing property on Λ. Since Λ is locally maximal in U, the shadowing point $y\in \mathrm{\Lambda}$. □
are ${C}^{1}$injectively immersed submanifolds of M.
Lemma 2.4 Let $p,q\in P(f)$ be hyperbolic saddles. If f has the ergodic shadowing property in a closed set Λ, then ${W}^{s}(p)\cap {W}^{u}(q)\ne \mathrm{\varnothing}$, and ${W}^{u}(p)\cap {W}^{s}(q)\ne \mathrm{\varnothing}$.
Proof Let $p,q\in P(f)$ be hyperbolic saddles, and let U be a locally maximal neighborhood of Λ. Suppose that f has the ergodic shadowing property in a locally maximal Λ. Since p and q are hyperbolic, there are $\u03f5(p)>0$ and $\u03f5(q)>0$ as in the above. Take $\u03f5=min\{\u03f5(p),\u03f5(q)\}/4$ and let $0<\delta \le \u03f5$ be the number of the ergodic shadowing property of f. For simplicity, we may assume that $f(p)=p$ and $f(q)=q$. Since f has the ergodic shadowing property in Λ, $f{}_{\mathrm{\Lambda}}$ is chain transitive. Then we can construct a finite δpseudo orbit form p to q as follows: ${x}_{0}=p$, ${x}_{n}=q$ ($n\ge 1$), and $d(f({x}_{i}),{x}_{i+1})<\delta $ for all $0<i<n1$. Put (i) ${x}_{i}={f}^{i}(p)$, for all $i\le 0$, and (ii) ${x}_{n+i}={f}^{i}(q)$ for all $i\ge 0$. Then we have the sequence $\xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}=\{\dots ,p,{x}_{1},{x}_{2},\dots ,{x}_{n},{x}_{n+1},\dots \}$. It is clearly a δergodic pseudo orbit of f. Since f has the ergodic shadowing property in Λ and locally maximal, by Lemma 2.2, f has the finite shadowing property on Λ and so, by [[9], Lemma 1.1.1], f has the shadowing property in Λ. By the shadowing property in Λ, we can show that $Orb(y)\subset {W}^{u}(p)\cap {W}^{s}(q)$, and so ${W}^{u}(p)\cap {W}^{s}(q)\ne \mathrm{\varnothing}$. The other case is similar. □
A diffeomorphism f is KupkaSmale if their periodic points of f are hyperbolic and if $p,q\in P(f)$, then ${W}^{s}(p)$ is transversal to ${W}^{u}(q)$. Then it is ${C}^{1}$residual in $Diff(M)$. Denote by $\mathcal{KS}(M)$ the set of all KupkaSmale diffeomorphisms. The following was proved by [10].
Lemma 2.5 [[10], Lemma 2.4]
Let Λ be locally maximal in U, and let $\mathcal{U}(f)$ be given. If for any $g\in \mathcal{U}(f)$, $p\in {\mathrm{\Lambda}}_{g}(U)\cap P(g)$ is not hyperbolic, then there is ${g}_{1}\in \mathcal{U}(f)$ such that ${g}_{1}$ has two hyperbolic periodic points $p,q\in {\mathrm{\Lambda}}_{{g}_{1}}(U)$ with different indices.
Denote by $\mathcal{F}(M)$ the set of $f\in Diff(M)$ such that there is a ${C}^{1}$ neighborhood $\mathcal{U}(f)$ of f such that, for any $g\in \mathcal{U}(f)$, every $p\in P(g)$ is hyperbolic. In [11], Hayashi proved that $f\in \mathcal{F}(M)$ if and only if f satisfies both Axiom A and the nocycle condition. We say that f is the local star condition diffeomorphism if there exist a ${C}^{1}$neighborhood $\mathcal{U}(f)$ and a neighborhood U of Λ such that, for any $g\in \mathcal{U}(f)$, every $p\in {\mathrm{\Lambda}}_{g}(U)\cap P(g)$ is hyperbolic (see [12]). Denote by $\mathcal{F}(\mathrm{\Lambda})$ the set of all local star diffeomorphisms. Note that there are a ${C}^{1}$neighborhood $\mathcal{U}(f)$ and a neighborhood U of p such that, for all $g\in \mathcal{U}(f)$, there is a unique hyperbolic periodic point ${p}_{g}\in U$ of g with the same period as p and $index({p}_{g})=index(p)$. Here $index(p)=dim{E}_{p}^{s}$, and the point ${p}_{g}$ is called the continuation of p.
Lemma 2.6 [[13], Lemma 2.2]
There is a residual set ${\mathcal{G}}_{1}\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_{1}$, if for any ${C}^{1}$neighborhood $\mathcal{U}(f)$ of f, there exists $g\in \mathcal{U}(f)$ such that two hyperbolic periodic points ${p}_{g},{q}_{g}\in P(g)$ with $index({p}_{g})\ne index({q}_{g})$, then f has two hyperbolic periodic points $p,q\in P(f)$ with $index(p)\ne index(q)$.
Proof Let $f\in {\mathcal{G}}_{2}={\mathcal{G}}_{1}\cap \mathcal{KS}(M)$, and let $p,q\in \mathrm{\Lambda}\cap P(f)$ be hyperbolic saddles. Suppose that f has the ergodic shadowing property in a locally maximal Λ. Then by Lemma 2.4 ${W}^{s}(p)\cap {W}^{u}(q)\ne \mathrm{\varnothing}$ and ${W}^{u}(p)\cap {W}^{s}(q)\ne \mathrm{\varnothing}$. Since $f\in {\mathcal{G}}_{2}$, ${W}^{s}(p)\u22d4{W}^{u}(q)\ne \mathrm{\varnothing}$ and ${W}^{u}(p)\u22d4{W}^{s}(q)\ne \mathrm{\varnothing}$. This means that $p\sim q$ and so $index(p)=index(q)$. □
Let p be a periodic point of f. For $0<\delta <1$, we say that p has a δweak eigenvalue if $D{f}^{\pi (p)}(p)$ has an eigenvalue λ such that ${(1\delta )}^{\pi (p)}<\lambda <{(1+\delta )}^{\pi (p)}$. We say that a periodic point has a real spectrum if all of its eigenvalues are real and simple spectrum if all its eigenvalues have multiplicity one. Denote by ${P}_{h}(f)$ the set of all hyperbolic periodic points of f.
Lemma 2.8 [[14], Lemma 5.1]
There is a residual set ${\mathcal{G}}_{3}\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_{3}$:

For any$\delta >0$, if for any${C}^{1}$neighborhood$\mathcal{U}(f)$of f there exist$g\in \mathcal{U}(f)$and${p}_{g}\in {P}_{h}(g)$with a δweak eigenvalue, then there is$p\in {P}_{h}(f)$with a 2δweak eigenvalue.

For any$\delta >0$, if$q\in {P}_{h}(f)$with a δweak eigenvalue and a real spectrum, then there is$p\in {P}_{h}(f)$with a δweak eigenvalue with a simple real spectrum.
Lemma 2.9 There is a residual set ${\mathcal{G}}_{4}\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_{4}$, if f has the ergodic shadowing property in a locally maximal Λ, then there exists $\eta >0$ such that, for any $q\in \mathrm{\Lambda}\cap {P}_{h}(f)$, q has no ηweak eigenvalues.
Proof Let $f\in {\mathcal{G}}_{4}={\mathcal{G}}_{2}\cap {\mathcal{G}}_{3}$ have the ergodic shadowing property in a locally maximal Λ. We will derive a contradiction. Suppose that, for any $\eta >0$, there is $q\in \mathrm{\Lambda}\cap {P}_{h}(f)$ such that q has an ηweak eigenvalue. By Franks’ lemma, there is g ${C}^{1}$close to f such that p is not hyperbolic. By Franks’ lemma and Lemma 2.5, there is h ${C}^{1}$nearby g and ${C}^{1}$close to f such that h has tow hyperbolic periodic points ${q}_{h}$, ${\gamma}_{h}$ with different indices. Since $f\in {\mathcal{G}}_{1}$, and it is locally maximal, by Lemma 2.6 f has two hyperbolic periodic points q, γ in Λ. Since f has the ergodic shadowing property in Λ, this is a contradiction by Lemma 2.7. □
Proposition 2.10 There is a residual set ${\mathcal{G}}_{4}\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_{4}$, if f has the ergodic shadowing property in Λ, then $f\in \mathcal{F}(\mathrm{\Lambda})$.
Proof Let $f\in {\mathcal{G}}_{4}$ have the ergodic shadowing property in a locally maximal Λ. Suppose by contradiction that $f\notin \mathcal{F}(\mathrm{\Lambda})$. Then there are g ${C}^{1}$close to f and $q\in P(g)$ such that q has a ηweak eigenvalue. Then by Lemma 2.9, we get a contradiction. Thus $f\in \mathcal{F}(\mathrm{\Lambda})$. □
Proposition 2.11 [[15], Proposition A]
where $\pi (p)$ is the period of p.
Remark 2.12 By Pugh’s closing lemma, there is a residual set ${\mathcal{G}}_{6}\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_{6}$, if $f{}_{\mathrm{\Lambda}}$ is transitive, then there is a periodic orbit ${p}_{n}$ such that $Orb({p}_{n})\to \mathrm{\Lambda}$ in Hausdorff metric.
Lemma 2.13 [[16], Theorem 3.8]
There is residual set ${\mathcal{G}}_{5}\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_{5}$, for any ergodic measure μ of f, there is a sequence of the periodic point ${p}_{n}$ such that ${\mu}_{{p}_{n}}\to \mu $ in weak^{∗} topology and $Orb({p}_{n})\to Supp(\mu )$ in Hausdorff metric.
The following was proved by Mañé [17]. Denote by $\mathcal{M}(f{}_{\mathrm{\Lambda}})$ the set of invariant probabilities on the Borel σalgebra of Λ endowed with the weak^{∗} topology.
for every ergodic $\mu \in \mathcal{M}({f}^{m}{}_{\mathrm{\Lambda}})$, then E is contracting.
By Lemma 2.14, E is contracting. Similarly, we can show that F is expanding. □
Corollary 2.15 For ${C}^{1}$generic f, if f has the ergodic shadowing property, then f is transitive Anosov.
Declarations
Acknowledgements
This work 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).
Authors’ Affiliations
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