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The ergodic shadowing property from the robust and generic view point
Advances in Difference Equations volume 2014, Article number: 170 (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.
1 Introduction
Let M be a closed ${C}^{\mathrm{\infty}}$ manifold, and let $Diff(M)$ 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 $\parallel \cdot \parallel $ on the tangent bundle TM. Let $f\in Diff(M)$. For $\delta >0$, a sequence of points ${\{{x}_{i}\}}_{i=a}^{b}$ ($\mathrm{\infty}\le a<b\le \mathrm{\infty}$) in M is called a δpseudo orbit of f if $d(f({x}_{i}),{x}_{i+1})<\delta $ for all $a\le i\le b1$. For given $x,y\in M$, we write $x\u21ddy$ if for any $\delta >0$, there is a δpseudo orbit ${\{{x}_{i}\}}_{i=a}^{b}$ ($a<b$) of f such that ${x}_{a}=x$ and ${x}_{b}=y$. Let Λ be a closed finvariant set. We say that f has the shadowing property in Λ if for every $\u03f5>0$ there is $\delta >0$ such that, for any δpseudo orbit ${\{{x}_{i}\}}_{i=a}^{b}\subset \mathrm{\Lambda}$ of f ($\mathrm{\infty}\le a<b\le \mathrm{\infty}$), there is a point $y\in \mathrm{\Lambda}$ such that $d({f}^{i}(y),{x}_{i})<\u03f5$ for all $a\le i\le b1$. If $\mathrm{\Lambda}=M$, then f has the shadowing property. The shadowing property usually plays an important role in the investigation of stability theory and ergodic theory. For instance, Sakai [1] proved that if f has the ${C}^{1}$robustly shadowing property, then f is structurally stable. Now we introduce the notion of the ergodic shadowing property which was introduced and studied by [2]. Lee has shown in [3] that if f belongs to the ${C}^{1}$interior of the set of all diffeomorphisms having the ergodic shadowing property, then it is structurally stable diffeomorphisms. In [4], Lee showed that if f is local star condition and has the ergodic shadowing property on the homoclinic class, then it is hyperbolic. For any $\delta >0$, a sequence $\xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}$ is a δergodic pseudo orbit of f if for $N{p}_{n}^{+}(\xi ,f,\delta )=\{i:d(f({x}_{i}),{x}_{i+1})\ge \delta \}\cap \{0,1,\dots ,n1\}$, and $N{p}_{n}^{}(\xi ,f,\delta )=\{i:d({f}^{1}({x}_{i}),{x}_{i1})\ge \delta \}\cap \{n+1,\dots ,1,0\}$
Here #A is the number of elements of the set A. We say that f has the ergodic shadowing property in Λ (or $f{}_{\mathrm{\Lambda}}$ has ergodic shadowing) if for any $\u03f5>0$, there is a $\delta >0$ such that every δergodic pseudo orbit $\xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda}$ of f there is a point $z\in \mathrm{\Lambda}$ such that, for $N{s}_{n}^{+}(\xi ,f,z,\u03f5)=\{i:d({f}^{i}(z),{x}_{i})\ge \u03f5\}\cap \{0,1,\dots ,n1\}$, and $N{s}_{n}^{}(\xi ,f,z,\u03f5)=\{i:d({f}^{i}(z),{x}_{i})\ge \u03f5\}\cap \{n+1,\dots ,1,0\}$,
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.
We say that Λ is locally maximal if there is a compact neighborhood U of Λ such that
Now, we introduce the notion of the ${C}^{1}$stably ergodic shadowing property in a closed set.
Definition 1.1 Let Λ be a closed finvariant set. We say that f has the ${C}^{1}$stably ergodic shadowing property in Λ if

(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.
We say that Λ is hyperbolic if the tangent bundle ${T}_{\mathrm{\Lambda}}M$ has a Dfinvariant splitting ${E}^{s}\oplus {E}^{u}$ and there exist constants $C>0$ and $0<\lambda <1$ such that
for all $x\in \mathrm{\Lambda}$ 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 $y\in \mathrm{\Lambda}$. □
Let $p\in P(f)$ be a hyperbolic saddle with period $\pi (p)>0$. Then there are the local stable manifold ${W}_{\u03f5}^{s}(p)$ and the local unstable manifold ${W}_{\u03f5}^{u}(p)$ of p for some $\u03f5=\u03f5(p)>0$. It is easily seen that if $d({f}^{n}(x),{f}^{n}(p))\le \u03f5$ for all $n\ge 0$, then $x\in {W}_{\u03f5}^{s}(p)$, and if $d({f}^{n}(x),{f}^{n}(p))\le \u03f5$ for all $n\le 0$, then $x\in {W}_{\u03f5}^{u}(p)$. The stable manifold ${W}^{s}(p)$ and the unstable manifold ${W}^{u}(p)$ defined as following. It is well known that if p is a hyperbolic periodic point of f with period k, then the sets
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 $index(p)\ne index(q)$.
Lemma 2.7 There is a residual set ${\mathcal{G}}_{2}\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_{2}$, if f has the ergodic shadowing property in a locally maximal Λ, then for any $p,q\in \mathrm{\Lambda}\cap P(f)$
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]
Let $f\in {\mathcal{G}}_{4}$, and let Λ be locally maximal. If f has the ergodic shadowing property in Λ, then there are $m>0$, $C\ge 1$ and $\lambda \in (0,1)$ such that, for any $p\in \mathrm{\Lambda}\cap P(f)$ with $\pi (q)>m$, we have
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.
Lemma 2.14 Let $\mathrm{\Lambda}\subset M$ be a closed finvariant set of f and $E\subset {T}_{\mathrm{\Lambda}}M$ be a continuous invariant subbundle. If there is $m>0$ such that
for every ergodic $\mu \in \mathcal{M}({f}^{m}{}_{\mathrm{\Lambda}})$, then E is contracting.
Proof of Theorem 1.4 Let $f\in {\mathcal{G}}_{4}\cap {\mathcal{G}}_{5}\cap {\mathcal{G}}_{6}$ have the ergodic shadowing property in a locally maximal Λ. Then by Proposition 2.11, we know that Λ admits a dominated splitting ${T}_{\mathrm{\Lambda}}M=E\oplus F$. Since f has the ergodic shadowing property in Λ, by Lemma 2.1, Remark 2.12, and Lemma 2.13, there is a sequence of periodic points such that $Orb({p}_{n})\to Supp(\mu )=\mathrm{\Lambda}$ in the Hausdorff metric. By Proposition 2.11, we have
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.
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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).
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Keywords
 ergodic shadowing
 shadowing
 locally maximal
 generic
 Anosov