Generic diffeomorphisms with weak limit shadowing

Abstract

In this paper, we show that if a $C^1$-generic diffeomorphism has the weak limit shadowing property on the chain recurrent set, then the diffemorphism satisfies Axiom A and the no-cycle condition.

MSC:37C50, 37D30.

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 $f\in Diff(M)$ and Λ be a closed f-invariant set. 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 δ-pseudo orbit of f if $d(f(x_i),x_{i+1})<\delta$ for all $a\le i\le b-1$.

We say that f has the shadowing property on Λ if for every $ϵ>0$, there is $\delta >0$ such that for any δ-pseudo orbit ${\{x_i\}}_{i=a}^b\subset \mathrm{\Lambda }$ of f ($-\mathrm{\infty }\le a<b\le \mathrm{\infty }$), there is a point $y\in M$ such that $d(f^i(y),x_i)<ϵ$ for all $a\le i\le b-1$. In the dynamical systems, the shadowing theory is a very useful notion. In fact, it deals with the stability theorem (see [1]). For instance, Robinson [2] proved that if a diffeomorphism f is structurally stable, then it has the shadowing property. In [3] Sakai showed that f belongs to the $C^1$-interior of the shadowing property if and only if f is structurally stable. In this paper, we deal with another shadowing property, that is, the weak limit shadowing property which was studied by [4].

We say that f has the weak limit shadowing property on Λ (or Λ is weak limit shadowable for f) if there exists a $\delta >0$ with the following property: if a sequence ${\{x_i\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda }$ is a δ-pseudo orbit of f, for which relations $d(f(x_i),x_{i+1})\to 0$ as $i\to +\mathrm{\infty }$ and $d(f^{-1}(x_{i+1}),x_i)\to 0$ as $i\to -\mathrm{\infty }$ hold, then there is a point $y\in M$ such that $d(f^i(y),x_i)\to 0$ as $i\to \pm \mathrm{\infty }$. Note that if f has the limit shadowing property, then f has the weak limit shadowing property. But the converse is not true (see [[4], Example 4]). Denote by $P(f)$ the set of periodic points of f. Then $P(f)\subset \mathrm{\Omega }(f)\subset \mathcal{R}(f)$, where $\mathrm{\Omega }(f)$ is the set of non-wandering points of f, and $\mathcal{R}(f)$ is the set of chain recurrent points of f. Note that if f satisfies Axiom A and the no-cycle condition, then $\mathrm{\Omega }(f)=\mathcal{R}(f)$. We say that f has the s-limit shadowing property on Λ if for any $ϵ>0$, there is a $\delta >0$ such that for any δ-limit pseudo orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda }$, there is a point $y\in M$ such that $d(f^i(y),x_i)<ϵ$ for all $i\in \mathbb{Z}$, and $d(f^i(y),x_i)\to 0$ as $i\to \pm \mathrm{\infty }$. Clearly, the weak limit shadowing property is a weak notion of the s-limit shadowing property. We say that Λ is hyperbolic if the tangent bundle $T_{\mathrm{\Lambda }}M$ has a Df-invariant splitting $E^s\oplus E^u$ and there exist constants $C>0$ and $0<\lambda <1$ such that

$$\parallel D_x{f}^n{|}_{E_x^s}\parallel \le C{\lambda }^n\text{and}\parallel D_x{f}^{-n}{|}_{E_x^u}\parallel \le C{\lambda }^n$$

for all $x\in \mathrm{\Lambda }$ and $n\ge 0$. If $\mathrm{\Lambda }=M$, then f is Anosov. Very recently, Sakai [5] showed that if a $C^1$-generic diffeomorphism f has the s-limit shadowing property on $\mathcal{R}(f)$, then f satisfies Axiom A and the no-cycle condition. The result is motivation for this study. The main theorem of the paper is as follows.

Theorem 1.1 For $C^1$-generic f, if f has the weak limit shadowing property on $\mathcal{R}(f)$, then f satisfies Axiom A and the no-cycle condition.

2 Proof of Theorem 1.1

Let M be as before and $f\in Diff(M)$. Let $p\in P(f)$ be a hyperbolic saddle with period $\pi (p)>0$. The stable manifold $W^s(p)$ and the unstable manifold $W^u(p)$ are defined as follows. It is well known that if p is a hyperbolic periodic point of f with a period k, then the sets

$$\begin{array}{c}{W}^s(p)=\{x\in M:f^{kn}(x)\to p\text{ as }n\to \mathrm{\infty }\}\text{and }\hfill \\ W^u(p)=\{x\in M:f^{-kn}(x)\to p\text{ as }n\to \mathrm{\infty }\}\hfill \end{array}$$

are $C^1$-injectively immersed submanifolds of M. Let $p,q\in P(f)$ be saddles. Let $P(f)$ be the set of periodic points of f. Denote by ${\mathcal{O}}_f(p)$ the periodic f-orbit of $p\in P(f)$. We denote $p\sim q$ if the intersections $W^s({\mathcal{O}}_f(p))⋔W^u({\mathcal{O}}_f(q))\ne \mathrm{\varnothing }$ and $W^u({\mathcal{O}}_f(p))⋔W^s({\mathcal{O}}_f(q))\ne \mathrm{\varnothing }$. Then we know that if $p\sim q$, then $index(p)=index(q)$. Here $index(p)$ is the dimension of the stable manifold of p, that is, $dim{W}^s(p)$.

Proposition 2.1 There is a residual set ${\mathcal{G}}_1\subset Diff(M)$ such that for any $f\in \mathcal{G}$, if $f{|}_{\mathcal{R}(f)}$ has the weak limit shadowing property, then for any saddles $p,q\in P(f)$, $index(p)=index(q)$.

To prove Proposition 2.1, we need the following lemma.

Lemma 2.2 Let $p,q\in P(f)$ be saddles. If f has the weak limit shadowing property on $\mathcal{R}(f)$, then $W^s({\mathcal{O}}_f(p))\cap W^u({\mathcal{O}}_f(q))\ne \mathrm{\varnothing }$.

Proof Suppose that f has the weak limit shadowing property on $\mathcal{R}(f)$. For any saddles $p,q\in P(f)$, we show that $W^u({\mathcal{O}}_f(p))\cap W^s({\mathcal{O}}_f(q))\ne \mathrm{\varnothing }$. For the sake of simplicity, we may assume that $f(p)=p$ and $f(q)=q$. Let $\delta >0$ be the number of the weak limit shadowing property of f such that $d(p,q)<\delta$. We construct δ-limit pseudo orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathcal{R}(f)$ as follows. (i) $x_0=p$, (ii) $x_{-i}=f^{-i}(p)$ for all $i>0$, (iii) $x_i=f^i(q)$ for all $i\ge 1$. Then the δ-limit pseudo orbit

$$\xi =\{\dots ,x_{-1},p=(x_0),q(=x_1),x_2,\dots \}=\{\dots ,p,p,q,q,\dots \},$$

and it is clear that $\xi \subset \mathcal{R}(f)$. Since f has the weak shadowing property on $\mathcal{R}(f)$, there is a point $y\in M$ such that $d(f^i(y),x_i)\to 0$ as $i\to \pm \mathrm{\infty }$. Then $f^i(y)\to p$ as $i\to -\mathrm{\infty }$ and $f^{1+i}(y)\to q$ as $i\to \mathrm{\infty }$. Hence, $y\in W^s(p)$ and $f(y)\in W^s(q)$. Thus, $W^u(p)\cap W^s(q)\ne \mathrm{\varnothing }$. □

The following is called the Kupka-Smale theorem.

Lemma 2.3 There is a residual set ${\mathcal{G}}_1\subset Diff(M)$ such that for any $f\in \mathcal{G}$, every periodic point is hyperbolic, and the stable manifolds and the unstable manifolds of periodic points are all transverse.

Proof of Proposition 2.1 Let $f\in {\mathcal{G}}_1$, and let $p,q\in P(f)$ be saddles. Suppose that f has the weak limit shadowing property on $\mathcal{R}(f)$. Let $\delta >0$ be the number of the weak limit shadowing property of f such that $d(p,q)<\delta$. Then we will drive a contradiction, we may assume that $index(p)\ne index(q)$. Then we know that $dim{W}^s(p)+dim{W}^u(q)<dimM$ or $dim{W}^u(p)+dim{W}^s(q)<dimM$. In this proof, we consider that $dim{W}^s(p)+dim{W}^u(q)<dimM$ (the other case is similar). Since $f\in {\mathcal{G}}_1$, $W^s(p)\cap W^u(q)=\mathrm{\varnothing }$. Since f has the weak limit shadowing property on $\mathcal{R}(f)$, by Lemma 2.2, $W^s(p)\cap W^u(q)\ne \mathrm{\varnothing }$. This is a contradiction. □

Let $p\in P(f)$ be a hyperbolic saddle with a period $\pi (p)>0$. Then there are the local stable manifold $W_{ϵ}^s(p)$ and the unstable manifold $W_{ϵ}^u(p)$ of p for some $ϵ=ϵ(p)>0$. It is easily seen that if $d(f^n(x),f^n(p))\le ϵ$ for all $n\ge 0$, then $x\in W_{ϵ}^s(p)$, and if $d(f^n(x),f^n(p))\le ϵ$ for all $n\le 0$, then $x\in W_{ϵ}^u(p)$. The following lemma shows that if f has the s-limit shadowing property on $\mathcal{R}(f)$, then the numbers of sinks and sources are finite (see [[5], Lemma 2]). From the above facts, we show that if f has the weak limit shadowing property on $\mathcal{R}(f)$, then the numbers of sinks and sources are finite.

Lemma 2.4 Let f have the weak limit shadowing property on $\mathcal{R}(f)$, and let $\delta >0$ be the number of the weak limit shadowing property of f. For any saddle $q\in P(f)$, if $p\in P(f)$ is a sink or a source, then $d(p,q)\ge \delta$.

Proof We will derive a contradiction. Suppose that $q\in P(f)$ is a saddle and $p\in P(f)$ is a sink with $d(p,q)<\delta$. For the sake of simplicity, we may assume that $f(p)=p$, $f(q)=q$. Since q is a saddle, there is $ϵ(q)>0$ such that if for any $x\in M$, $d(f^i(x),f^i(q))\le ϵ(q)$ as $i\to \mathrm{\infty }$, then $x\in W_{ϵ(q)}^s(q)$, and if $x\in M$, $d(f^i(x),f^i(q))\le ϵ(q)$ as $i\to -\mathrm{\infty }$, then $x\in W_{ϵ(q)}^u(q)$. Then we may assume that $d(p,q)>ϵ(q)$. Then we construct a δ-limit pseudo orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathcal{R}(f)$ as follows. Put $x_{-i}=f^{-i}(p)$ for $i\ge 0$ and $x_i=f^i(q)$ for $i\ge 1$. Then $\xi ={\{x_i\}}_{i\in \mathbb{Z}}$ is clearly a δ-limit pseudo orbit of f, and $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathcal{R}(f)$. Since f has the weak limit shadowing property on $\mathcal{R}(f)$, there is a point $y\in M$ such that $d(f^i(y),x_i)\to 0$ as $i\to \pm \mathrm{\infty }$. Since p is a sink, $d(f^{-i}(y),x_{-i})=d(f^{-i}(y),p)\to 0$ as $i\to \mathrm{\infty }$. Then $y=p$. Since $d(f^i(y),x_i)=d(f^i(y),q)\to 0$ as $i\to \mathrm{\infty }$, there is $k>0$ such that $d(f^{k+i}(y),f^{k+i}(q))=d(f^{k+i}(y),q)\le ϵ(q)$ for $i\ge 0$. Then $f^k(y)\in W_{ϵ(q)}^s(q)$. Since $y=p$, we know that $d(p,q)\le ϵ(q)$. This is a contradiction. □

Let p be a periodic point of f, and let $0<\delta <1$. We say p has a δ-weak eigenvalue provided $D_p{f}^{\pi (p)}$ has an eigenvalue λ such that ${(1-\delta )}^{\pi (p)}<|\lambda |<{(1+\delta )}^{\pi (p)}$. We say that the periodic point has a real spectrum if all of its eigenvalues are real and a simple spectrum if all of its eigenvalues have multiplicity one. The following lemma will play a crucial role in our proof.

Lemma 2.5 [[6], Lemma 5.1]

There is a residual set ${\mathcal{G}}_2\subset Diff(M)$ such that for any $f\in {\mathcal{G}}_2$,

• (a) for any $\eta >0$, if for any $C^1$-neighborhood $\mathcal{U}(f)$ of f, there exist $g\in \mathcal{U}(f)$ and $p_g,q_g\in P(g)$ with the same period such that $d(p_g,q_g)<\eta$, then there exist $p,q\in P(f)$ with the same period such that $d(p,q)<\eta$;

• (b) for any $\eta >0$, if for any $C^1$-neighborhood $\mathcal{U}(f)$ of f, there exist $g\in \mathcal{U}(f)$ and $p_g\in P(g)$ with an η-weak eigenvalue, then there exist $p\in P(f)$ with a 2η-weak eigenvalue;

• (c) for any $\eta >0$, if $q\in P(f)$ with an η-weak eigenvalue and a real spectrum, then there exists $p\in P(f)$ with an η-weak eigenvalue with a simple real spectrum.

Lemma 2.6 [[7], 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 $\eta >0$, if for any $C^1$-neighborhood $\mathcal{U}(f)$ of f, there exist $g\in \mathcal{U}(f)$ and $p_g,q_g\in P(g)$ with the same period such that $d(p_g,q_g)<\eta$ with different indices, then there exist $p,q\in P(f)$ with the same period such that $d(p,q)<\eta$ with different indices.

The following so-called Franks lemma will play an essential role in our proof.

Lemma 2.7 Let $\mathcal{U}(f)$ be any given $C^1$-neighborhood of f. Then there exists $ϵ>0$ and a $C^1$-neighborhood ${\mathcal{U}}_0(f)\subset \mathcal{U}(f)$ of f such that for given $g\in {\mathcal{U}}_0(f)$, a finite set $\{x_1,x_2,\dots ,x_N\}$, a neighborhood U of $\{x_1,x_2,\dots ,x_N\}$ and linear maps $L_i:T_{x_i}M\to T_{g(x_i)}M$ satisfying $\parallel L_i-D_{x_i}g\parallel \le ϵ$ for all $1\le i\le N$, there exists $g^{\prime }\in \mathcal{U}(f)$ such that $g^{\prime }(x)=g(x)$ if $x\in \{x_1,x_2,\dots ,x_N\}\cup (M\setminus U)$ and $D_{x_i}{g}^{\prime }=L_i$ for all $1\le i\le N$.

If $p\in P(f)$ is hyperbolic, then for any $g\in \mathcal{U}(f)$, there is a unique hyperbolic periodic point $p_g\in P(g)$ nearby p such that $\pi (p_g)=\pi (p)$ and $index(p_g)=index(p)$, where $index=dim{W}^s(p)$. Such a $p_g$ is called the continuation of p.

Lemma 2.8 There is a residual set ${\mathcal{G}}_4\subset Diff(M)$ such that for any $f\in {\mathcal{G}}_4$, if f has the weak limit shadowing property on $\mathcal{R}(f)$, then there is $\eta >0$ such that f has no η-weak eigenvalue.

Proof Let $f\in {\mathcal{G}}_4={\mathcal{G}}_2\cap {\mathcal{G}}_3$. To derive a contradiction, we may assume that for any $\eta >0$, there is a hyperbolic periodic point $q_g$ of g ($C^1$-nearby f) such that $q_g$ has an η-weak eigenvalue and a simple real spectrum. Let $\delta >0$ be the number of the weak limit shadowing property of f such that $0<\eta <\delta /2$. For the sake of simplicity, we assume that $q_g$ is a fixed point. By Lemma 2.7, there is h $C^1$-close to g and h $C^1$-nearby f such that $q_h$ has 1 as an eigenvalue. By Lemma 2.7 and as in the proof of [[8], Lemma 2.4], we can construct an $h^l$ ($l>0$)-invariant small arc ${\mathcal{I}}_{q_h}$ of $h^l$ containing $q_h$ such that $d(p_h,r_h)<\eta$, where $p_h$, $r_h$ are the end points of ${\mathcal{I}}_{q_h}$, $h^l(p_h)=p_h$, $h^l(r_h)=r_h$ and $p_h$, $r_h$ are hyperbolic saddles and different indices. Since $f\in {\mathcal{G}}_4$, there exist $p,r\in P(f)$ with the same period such that $d(p,r)<\eta$ with different indices. Since f has the weak limit shadowing property on $\mathcal{R}(f)$, and by Lemma 2.4, $p,r\in P(f)$ are saddles. Since $d(p,r)<\delta$, by Proposition 2.1, we know that $index(p)=index(r)$. But, since $index(p)\ne index(r)$, this is a contradiction. □

Denote by $\mathcal{F}(M)$ the $C^1$-interior of the set of diffeomorphisms of M whose periodic points are all hyperbolic. In [9], Hayashi showed that if $f\in \mathcal{F}(M)$, then f satisfies Axiom A and the no-cycle condition. To prove Theorem 1.1, it is enough to show $f\in \mathcal{F}(M)$.

End of proof of Theorem 1.1 Let $f\in {\mathcal{G}}_4$, and let f have the weak limit shadowing property on $\mathcal{R}(f)$. If not, then $f\notin \mathcal{F}(M)$. There is g $C^1$-closed to f and a non-hyperbolic periodic point $p_g$ such that the point $p_g$ has an $\eta /2$ weak eigenvalue. Since $f\in {\mathcal{G}}_4$, there is $p\in P(f)$ such that p has an η-weak eigenvalue. By Lemma 2.8, this is a contradiction. □