Local Topological Stability for Diffeomorphisms

Abstract

Let $f:M\rightarrow M$ be a diffeomorphism of compact smooth Riemannian manifold M,  an let $\Lambda \subset M$ be a closed f-invariant set. We obtain conditions for $\Lambda$ to be topologically stable which is called $\Lambda$-topologically stable. Moreover, we prove that if f is $C^1$ robustly $\Lambda$-topologically stable then $\Lambda$ satisfies star condition for f. Then in the above, if a closed f-invariant set $\Lambda$ is chain transitive (or transitive) then it is hyperbolic for f.

1 Introduction

In this paper, we assume that M is a compact smooth Riemannian manifold with $\textrm{dim}M\ge 2$. Denote by $\textrm{Diff}(M)$ the space of diffeomorphisms of M endowed with the $C^1$-topology. Let d be the distance on M induced from a Riemannian metric $\Vert \cdot \Vert$ on the tangent bundle TM. A closed f-invariant set $\Lambda \subset M$ is hyperbolic for f if the tangent bundle $T_{\Lambda }M$ has a Df-invariant splitting $E^s\oplus E^u$ and there exist constants $C>0$ and $0<\lambda <1$ such that

$$\begin{aligned} \Vert Df^n_{|_{E_x^s}}\Vert \le C\lambda ^n\;\;\textrm{and}\;\;\Vert Df^{-n}_{|_{E_x^u}}\Vert \le C\lambda ^{n} \end{aligned}$$

for all $x\in \Lambda$ and $n\ge 0.$ If $\Lambda =M$ then we say that f is Anosov. For any homeomorphisms $f, g:M\rightarrow M$, we defined by the $C^0$ metric

$$\begin{aligned} d_0(f, g)=\sup \{x\in M: d(f(x), g(x)), d(f^{-1}(x), g^{-1}(x))\}. \end{aligned}$$

Walters [17, 18] studies topologically stable, that is, a diffeomorphism f is topologically stable if for any $\epsilon >0$, there is $\delta >0$ such that for any homeomorphism $g:M\rightarrow M$ with $d_0(f, g)<\delta$ there is a continuous map $h:M\rightarrow M$ such that $h\circ g=f\circ h$ and $d(h(x), x)<\epsilon$ for any $x\in M,$ where $d_0$ is the $C^0$ metric.

We say that a diffeomorphism f is expansive if there is $\delta >0$ such that for any $x, y\in M$, if $d(f^i(x), f^i(y))\le \delta$ for all $i\in {\mathbb {Z}}$ then $x=y$. Given $d>0$, a sequence $\{x_i\}_{i \in {\mathbb {Z}}}$ of points in M is called a d-pseudo orbit of f if the following is satisfied

$$\begin{aligned} d(f(x_k), x_{k+1})<d, \end{aligned}$$

for all $k \in {\mathbb {Z}}$. We say that a diffeomorphism f has the shadowing property if for any $\epsilon >0$ there is $d>0$ such that any d-pseudo orbit $\{x_i\}_{i \in {\mathbb {Z}}}$ of points in M of f there is a point $y\in M$ such that

$$\begin{aligned} d(f^i(y), x_i)<\epsilon \end{aligned}$$

for all $i\in {\mathbb {Z}}.$

Walters [18] proved that if an expansive diffeomorphism f has the shadowing property then f is topologically stable. Also, he [17] proved that if a diffeomorphism f is Anosov then f is topologically stable. Denote by P(f) the set of all periodic points of f. We say that a diffeomorphism f satisfies Axiom A if the non-wandering set $\Omega (f)$ is hyperbolic and it is the closure of P(f). For any diffeomorphisms $f, g:M\rightarrow M$, we defined by the $C^1$ metric

$$\begin{aligned} d_1(f, g)=d_0(f, g)+\max _{x\in M}\max (\Vert Df(x)-Dg(x)\Vert , \Vert Df^{-1}(x)-Dg^{-1}(x)\Vert ). \end{aligned}$$

A diffeomorphism f is $\Omega$-stable if there is a $C^1$ neighborhood ${\mathcal {U}}$ of f such that for any $g\in {\mathcal {U}}$ there is a homeomorphism $h:\Omega (f)\rightarrow \Omega (g)$ such that $f\circ h=h\circ g,$ where $\Omega (g)$ is the non-wandering set of g. Smale [16] and Palis [13] proved that a diffeomorphism f satisfies Axiom A and the no-cycle condition if and only if f is $\Omega$-stable. The following notion was introduced by Gan and Wen [5]. We say that a diffeomorphism f is star if there is a $C^1$ neighborhood ${\mathcal {U}}$ of f such that for any $g\in {\mathcal {U}}$, every $p\in P(g)$ is hyperbolic, where P(g) is the set of all periodic point of g. Denote by ${\mathcal {F}}(M)$ the set of all star diffeomorphisms. By Aoki [1] and Hayashi [6], if a diffeomorphism $f\in {\mathcal {F}}(M)$ then f is $\Omega$-stable.

We say that a diffeomorphism f is $\Omega$-topologically stable if for any $\epsilon >0$ there is $\delta >0$ such that for any homeomorphism $g:M\rightarrow M$ with $d_0(f, g)<\delta$ there is a continuous map $h:\Omega (g)\rightarrow \Omega (f)$ such that

• (a) $f\circ h=h\circ g$ on $\Omega (g),$ and

• (b) $d(h(x), x)<\epsilon$ for $x\in \Omega (g).$

Moriyasu [10] proved that if a diffeomorphism f belongs to the $C^1$ interior of the set of all topologically stable then it is structurally stable. Moreover, he proved that if a diffeomorphism f belongs to the $C^1$ interior of the set of all $\Omega$-topologically stable then it satisfies Axiom A and the no-cycle condition.

In this paper we generalize Moriyasu result. We say that $\Lambda \subseteq M$ is locally maximal with respect to some $f:M\rightarrow M$ if there is a compact neighborhood U (called an isolating block) such that

$$\begin{aligned} \Lambda =U_f=\bigcap _{n\in {\mathbb {Z}}}f^n(U). \end{aligned}$$

Every locally maximal invariant set is clearly a compact invariant set of f.

Definition 1.1

Let $f:M\rightarrow M$ be a diffeomorphism and $\Lambda$ be a locally maximal invariant set of f. We say that f is $\Lambda$-topologically stable if for any $\epsilon >0$ there are an isolating block U of $\Lambda$ and a $C^0$ neighborhood ${\mathcal {U}}$ of f such that for any $g\in {\mathcal {U}}$ there is a continuous map $h: U_g\rightarrow \Lambda$ such that $d(h(x),x)\le \epsilon$ for every $x\in U_g$ and $f\circ h=h\circ g$ on $U_g$, where $U_g=\bigcap _{n\in {\mathbb {Z}}}g^n(U)$ is the continuation of $\Lambda$.

Note that we introduce another definition of the topological stability of a set $\Lambda$ which is corresponding to the notion of the $C^0$ lower semistability of the germ of $\Lambda$(see [11]).

Definition 1.2

Let $f:M\rightarrow M$ be a diffeomorphism and $\Lambda$ be a locally maximal invariant set of f. We say that f is $C^1$ robustly $\Lambda$-topologically stable if there are an isolated block U of $\Lambda$ and a $C^1$ neighborhood ${\mathcal {U}}$ of f such that g is $U_g$-topologically stable, for every $g\in {\mathcal {U}}$.

A closed f-invariant set $\Lambda \subset M$ satisfies a local star condition for f (or f is local star on $\Lambda$) if there are an isolated block U of $\Lambda$ and a $C^1$ neighborhood ${\mathcal {U}}$ of f such that for any $g\in {\mathcal {U}},$ every $p\in U_g\cap P(g)$ is hyperbolic, where P(g) is the set of all periodic points of g. The notion is a generalization of star diffeomorpisms, because if $\Lambda =M$ then it satisfies star condition. For the notion, a remarkable result is Lee [8]. In [8], the author showed that if a transitive set $\Lambda$ satisfies a local star condition for f then $\Lambda$ is hyperbolic for f. From the local star condition, we will show the following Theorem.

Theorem A

Let $\Lambda \subset M$ be a closed f-invariant set for f. If a diffeomorphism f is $C^1$ robustly $\Lambda$-topologically stable, then $\Lambda$ satisfies a local star condition for f.

A closed f-invariant set ${\mathcal {C}}$ is called chain transitive if for any $\delta >0$ and $x, y\in {\mathcal {C}}$, there is a $\delta$-pseudo orbit $\{x_i\}_{i=0}^n(n\ge 1)\subset {\mathcal {C}}$ such that $x_0=x$ and $x_n=y.$ For any hyperbolic periodic point p,  we denote $\textrm{index}(p)=\textrm{dim}W^s(p),$ where $W^s(p)=\{x\in M: f^i(x)\rightarrow p$ for $i\rightarrow \infty \}$ which is called the stable manifold of x. A closed f-invariant set $\Lambda$ with $\textrm{index}(p)=i(p\in \Lambda \cap P(f))$ is robustly homogenous index if there are an isolating block U of $\Lambda$ and a $C^1$ neighborhood ${\mathcal {U}}$ of f such that for any $g\in {\mathcal {U}}$ every hyperbolic $q\in U_g\cap P(g)$ has $\textrm{index}(q)=i$. Lee [9] proved that if every periodic points in ${\mathcal {C}}$ is robustly homogenous index then the set is hyperbolic. In the paper, we consider the chain transitive set ${\mathcal {C}}$ under a type of a locally topological stability. We prove the following.

Theorem B

Let $f:M\rightarrow M$ be a diffeomorphism and ${\mathcal {C}}\subset M$ be a locally maximal chain transitive set. If f is $C^1$ robustly ${\mathcal {C}}$-topologically stable, then ${\mathcal {C}}$ is a hyperbolic set of f.

2 Proof of Theorem A

Let $f:M\rightarrow M$ be a diffeomorphism. A closed f-invariant set $\Lambda$ is called transitive if there is a point $x\in \Lambda$ such that $\omega (x)=\Lambda$. It is known that according to $C^1$ closing lemma, if $\Lambda$ is transitive and locally maximal, then for every isolating block U there is g $C^1$ close to f such that g has a periodic point in $U_g=\bigcap _{n\in {\mathbb {Z}}}g^n(U).$ Then, we have the following lemma.

Lemma 2.1

Let $\Lambda$ be a locally maximal invariant set of f. If a diffeomorphism f is $\Lambda$-topologically stable, then $\Lambda \cap P(f)\not =\emptyset .$

Proof

Let $\epsilon >0$ be given. For this $\epsilon$ take the isolating block U of $\Lambda$ and a $C^0$ neighborhood ${\mathcal {U}}$ of f from the $\Lambda$-topological stability of f. Take any $x\in \Lambda$. We have $\omega (x)\subset \Lambda$. By $C^0$ closing lemma, we can find $g\in {\mathcal {U}}$ and a periodic orbit $Orb(p_g)$ such that the orbit of $Orb(p_g)$ close to $\omega (x)$ (Hausdorff arbitrarily close). This implies $p_g\in U_g$. Now let $h:U_g\rightarrow \Lambda$ be the continuous map given in the definition of $\Lambda$-topological stability. It follows that $h(p_g)$ is well defined and belongs to $\Lambda$. Since $h\circ g=f\circ h$ on $U_g$,

$$\begin{aligned} h(g^n(p_g))=f^n(h(p_g)) \end{aligned}$$

for all $n\in {\mathbb {Z}}.$ So, if k is the period of $p_g$ with respect to g,

$$\begin{aligned} f^k(h(p_g))=h(g^k(p_g))=h(p_g), \end{aligned}$$

proving that $h(p_g)$ is the periodic point of f in $\Lambda$. $\square$

The following is called Franks’ lemma [4] which is a very useful lemma for the $C^1$ perturbation property.

Lemma 2.2

Let ${\mathcal {U}}(f)$ be any given $C^1$ neighborhood of f. Then there exist $\epsilon >0$ 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 $\{x_1, x_2, \cdots , x_N\}$, a neighborhood W of $\{x_1, x_2, \cdots , x_N\}$ and linear maps $L_i : T_{x_i}M\rightarrow T_{g(x_i)}M$ satisfying $\Vert L_i-D_{x_i}g\Vert \le \epsilon$ for all $1\le i\le N$, there exists ${\widehat{g}}\in {\mathcal {U}}(f)$ such that ${\widehat{g}}(x)=g(x)$ if $x\in \{x_1, x_2, \cdots , x_N\}\cup (M\setminus W)$ and $D_{x_i}{\widehat{g}}=L_i$ for all $1\le i\le N$.

According to Lemma 2.1, if f is $\Lambda$-topologically stable then there is a periodic point p which contained in $\Lambda$. From this we can see the following fact.

Proposition 2.3

Let $f:M\rightarrow M$ be a diffeomorphism and $\Lambda$ be a locally maximal invariant set of f. Suppose that f is $C^1$ robustly $\Lambda$-topologically stable. Then, there are an isolating block U of $\Lambda$ and a $C^1$ neighborhood ${\mathcal {U}}$ of f such that every $p\in U_g\cap P(g)$ is hyperbolic, for any $g\in {\mathcal {U}}$.

Proof

Let U be the isolating block of $\Lambda$ and ${\mathcal {U}}$ be the $C^1$ neighborhood of f given in the definition of $C^1$ robustly $\Lambda$-topological stability. Put ${\mathcal {U}}(f)={\mathcal {U}}$ in Lemma 2.2 to get $\epsilon$ and the neighborhood ${\mathcal {U}}_0(f)\subset {\mathcal {U}}$. Suppose that there are $g\in {\mathcal {U}}_0(f)$ and $p\in P(g)\cap U_g$ such that p is not hyperbolic for g. Since $p\in P(g)\cap U_g$ is not hyperbolic, there is an eigenvalue $\lambda$ of $D_pg^{\pi (p)}$ such that $|\lambda |=1,$ where $\pi (p)$ is the period of p. For simplicity, we may assume that $\pi (p)=1.$ Since p is not hyperbolic for g,  we assume that $T_pM=E^c_p\oplus E^s_p\oplus E^u_p$ is the $D_pg$-invariant splitting of $T_pM$, where $E^{\sigma }_p$, $\sigma =c,s,u$, are subspaces $T_pM$ corresponding to eigenvalues $\lambda$ of $D_pg$ for $|\lambda |=1, |\lambda |<1$ and $|\lambda |>1$, respectively.

If $\lambda \in {\mathbb {R}}$ then we consider $\lambda =1$. In Lemma 2.2 we put $N=1$, $x_1=p$, $W=B_\alpha (p)$ in a way that $W\subseteq U$ and $L=L_1:T_pM\rightarrow T_pM$ such that $L|_{E^*_p}=D_pg|_{E^*_p}+\epsilon I|_{E^*_p}$ for $*=s,u$ and $L|_{E^c_p}=I$. Then, $\Vert L-D_pg\Vert \le \epsilon$ and so, by Lemma 2.2, we can choose $\varphi ={\widehat{g}}\in {\mathcal {U}}(f)$ such that $\varphi (p)=g(p)$, $\varphi (x)=g(x)$ for $x\in M\setminus W$ and $D_p\varphi =L$. We can assume that the exponential map $exp_p:T_pM\rightarrow W$ is a well defined diffeomorphisms.

Take a non-zero vector $u\in E^c_p$ such that $\Vert u\Vert \le \alpha /2$. Then we have

$$\begin{aligned} \varphi (\textrm{exp}_p(u)) = \textrm{exp}_p \circ L\circ \textrm{exp}^{-1}_p (\textrm{exp}_p(u)) = \textrm{exp}_p(u). \end{aligned}$$

Denoting by $E^c_p(\alpha /4)$ the ball of radius $\alpha /4$, centered in $\overrightarrow{O}_p$ and inside $E^c_p$, we have an invariant small arc ${\mathfrak {I}}_{p}\subset B_{\alpha }(p)\cap \textrm{exp}_p(E^c_p(\alpha /4))$ with center at p which satisfies the following:

• (1) ${\mathfrak {I}}_{p}\subset U_{\varphi }=\bigcap _{n\in {\mathbb {Z}}}\varphi ^n(U)$;

• (2) $\varphi ({\mathfrak {I}}_p)={\mathfrak {I}}_p$;

• (3) $\varphi |_{{\mathfrak {I}}_p}:{\mathfrak {I}}_p\rightarrow {\mathfrak {I}}_p$ is the identity map;

• (4) ${\mathfrak {I}}_p$ is a normally hyperbolic set of $\varphi$ (see proof of Proposition A p. 730 in [19]).

Since f is $C^1$ robustly $\Lambda$-topologically stable, $\varphi$ is $U_\varphi$-topologically stable. However, we shall prove that $\varphi$ is not $U_\varphi$-topologically stable as follows.

Let $diam({\mathfrak {J}}_p)$ be the diameter of ${\mathfrak {J}}_p$. By Item (4) above we can choose $0<\rho <diam({\mathfrak {J}}_p)$ so that

$$\begin{aligned} {\mathfrak {J}}_p=\bigcap _{n\in {\mathbb {Z}}}\varphi ^n(O) \end{aligned}$$

(1)

where O is the $\rho$-ball centered at ${\mathfrak {J}}_p$. Choose $\delta$ from the $U_\varphi$-topological stability of $\varphi$ for $\rho /4$. Now take $\phi$ $C^1$ $\delta$-close to $\varphi$ so that $\phi ({\mathfrak {J}}_p)={\mathfrak {J}}_p$ and the dynamics of $\phi |_{{\mathfrak {J}}_p}$ is Pole North-South one namely we identify ${\mathfrak {J}}_p=[0,1]$ with $\phi (0)=0$, $\phi (1)=1$ and $\phi ^n(y)\rightarrow 0$ or 1 as $n\rightarrow \infty$ respectively for $y\in ]0,1[$.

It follows that there is a continuous map $h:U_\phi \rightarrow U_\varphi$ so that $d(h(y),y)<\rho /4$ and $\varphi \circ h=h\circ \phi$. Since $\phi ({\mathfrak {J}}_p)={\mathfrak {J}}_p$, we have ${\mathfrak {J}}_p\subseteq U_\phi$. Therefore h is defined on ${\mathfrak {J}}_p$.

Note if $y\in {\mathfrak {J}}_p$ then $\varphi (h(y))=h(\phi (y))\in h({\mathfrak {J}}_p)$ proving that $h({\mathfrak {J}}_p)$ is $\varphi$-invariant. By (1) we get $h({\mathfrak {J}}_p)\subseteq {\mathfrak {J}}_p$.

If $y\in ]0,1[$ then $\phi ^n(y)\rightarrow 1$ as $n\rightarrow \infty$ thus $h(y)=\varphi ^n(h(y))=h(\phi ^n(y))\rightarrow h(1)$ proving $h(y)=1$ for every $y\in ]0,1[$. Then, $h(y)=h(1)$ for every y by continuity. It follows that

$$\begin{aligned} d(y,w)\le d(h(y),y)+d(h(y),h(w))+d(h(w),w)<\rho /2<diam({\mathfrak {J}}_p),\quad \forall y,w\in {\mathfrak {J}}_p. \end{aligned}$$

This is a contradiction proving the result.

If the eigenvalue $\lambda \in {\mathbb {C}}$, then to avoid notational complexity, we consider only the case $g(p)=p.$ As in the above case, there are $\alpha >0$ and $g_1$ $C^1$ close to g $(h\in {\mathcal {U}})$ such that $g_1(p)=g(p)=p$ and

$$\begin{aligned} g_1(x)=\textrm{exp}_p\circ D_pg\circ \textrm{exp}_p^{-1}(x)\ \text{ if }\ x\in B_{\alpha }(p). \end{aligned}$$

Then there is $k>0$ such that $D_pg_1^k(v)=v$ for any non-zero vector $v\in E^c_p.$ As in the above argument, we can get a contradiction. $\square$

Denote by ${\mathcal {F}}(\Lambda )$ the set of all diffeomorphisms satisfying the local star condition on $\Lambda$. To prove Theorem A, it is enough to show that a diffeomorphism $f\in {\mathcal {F}}(\Lambda ).$

Proof of Theorem A

Since a diffeomorphism f is $\Lambda$-topologically stable, according to Lemma 2.1, $P(f)\cap \Lambda \not =\emptyset .$ Since f is $C^1$ robustly $\Lambda$-topologically stable, by Proposition 2.3, $f\in {\mathcal {F}}(\Lambda ).$ This ends proof of Theorem A. $\square$ .

3 Proof of Theorem B

Let M be as before and let $f\in \textrm{Diff}(M).$

Lemma 3.1

Let $f:M\rightarrow M$ be a diffeomorphism and let $\Lambda \subset M$ be a closed f-invariant set. Let $k>0$ be an integer and $\epsilon>0, \eta >0$ be given. Then for any sequence $\{x_0, x_1, \ldots , x_k\}\subset \Lambda$ with $d(f(x_i), x_{i+1})<\epsilon$ for $i\in \{0, 1,\ldots , k-1\},$ there exists a sequence $\{y_0, y_1, \ldots , y_k\}\subset \Lambda$ such that

• (i) $d(x_i, y_i)<\eta$, for $i\in \{0, 1, \ldots , k\},$

• (ii) $d(f(y_i), y_{i+1})<2\epsilon$, for $i\in \{0, 1, \ldots , k-1\},$ and

• (iii) $y_i\not =y_j (i\not =j),$ for $0\le i, j\le k.$

Proof

The proof is similar to [2, Lemma 2.4.10]. $\square$

For any $\delta >0$, a sequence $\{x_i\}_{i\in {\mathbb {Z}}}$ is said to be $\delta$ pseudo orbit of f if $d(f(x_i), x_{i+1})<\delta$ for all $i\in {\mathbb {Z}}.$ We say that a diffeomorphism f has the shadowing property on $\Lambda$ if for any $\epsilon >0$ there is $\delta >0$ such that for any $\delta$ pseudo orbit $\{x_i\}_{i\in {\mathbb {Z}}}\subset \Lambda$ we can take a point $z\in M$ satisfying $d(f^i(z), x_i)<\epsilon$ for all $i\in {\mathbb {Z}}.$

Theorem 3.2

Let $\Lambda \subset M$ be a closed f-invariant set. If a diffeomorphism f is $\Lambda$-topological stable then f has the shadowing property on $\Lambda .$

Proof

For any $\epsilon >0$, let U be a locally maximal neighborhood of $\Lambda$ and let $\delta >0$ be as corresponding to the definition of $\Lambda$-topologically stable. We assume that $\{x_0, x_1, \ldots , x_k\}\subset \Lambda$ be choosen such that $d(f(x_i), x_{i+1})<\delta /{4\pi }$ for $i=\{0, 1, \ldots , k-1\}.$ According to Lemma 3.1, there is $\{y_0, y_1, \ldots , y_k\}\subset \Lambda$ such that

• (i) $d(x_i, y_i)<\epsilon$, for $i\in \{0, 1, \ldots , k\}$,

• (ii) $d(f(y_i), y_{i+1})<\delta /{2\pi }$ for $i\in \{0, 1, \ldots , k-1\}$,

• (iii) $y_i\not = y_j (i\not =j)$ for $0\le i, j\le k$, and

• (iv) $f(y_i)\not = f(y_j)(i\not =j)$ for $0\le i, j\le k.$

According to [12, Lemma 13], there is a homeomorphism $\zeta :M\rightarrow M$ such that $d(\zeta (x), x)<\delta$ for $x\in M$ and $\zeta \circ f(y_i)=y_{i+1}$ for $i\in \{0, 1, \ldots , k-1\}.$ Let $g=\zeta \circ f.$ Then we have $d(g(x), f(x))<\delta$ for $x\in M$ and $g(y_i)=y_{i+1}$ for $i\in \{0, 1, \ldots , k-1\}.$ Since f is $\Lambda$-topologically stable, there are a closed invariant set $U_g=\bigcap _{n\in {\mathbb {Z}}}g^n(U)$ and a continuous map $h:U_g\rightarrow \Lambda$ such that $d(h(x), x)<\epsilon$ for $x\in U_g$ and $h\circ g=f\circ h$ on $U_g.$ Then we have

$$\begin{aligned} d(f^i\circ h(y_0), x_i)&=d(h\circ g^i(y_0), x_i)\\ {}&=d(h(y_i), x_i)\le d(h(y_i), y_i)+d(y_i, x_i)\\ {}&<\epsilon +\epsilon =2\epsilon , \end{aligned}$$

for $i\in \{0, 1, \ldots , k\}.$ Thus for each $\{x_i\}_{i=0}^k\subset \Lambda$ with $d(f(x_i), x_{i+k})<\delta /4\pi$ for $i\in \{0, 1, \ldots , k-1\},$ there is $y\in \Lambda$ such that $d(f^i(y), x_i)<2\epsilon$, for $i\in \{0, 1, \ldots , k-1\}.$ Since $\Lambda \subset M$ is a closed and invariant set for f, by [14, Lemma 1.1.1] f has the shadowing property on $\Lambda .$ $\square$

We say that a diffeomorpism f has the $C^1$ robustly shadowing property on $\Lambda$ if there are a $C^1$ neighborhood ${\mathcal {U}}(f)$ of f and an isolating block U of $\Lambda$ such that for any $g\in {\mathcal {U}}(f)$, g has the shadowing property on $U_g.$

Proof of Theorem B

Since f is $C^1$ robustly ${\mathcal {C}}$-topologically stable, there are a $C^1$ neighborhood ${\mathcal {U}}(f)$ of f and an isolated block U of ${\mathcal {C}}$ such that for any $g\in {\mathcal {U}}$, g is $U_g$-topologically stable, where $U_g=\bigcap _{n\in {\mathbb {Z}}}g^n(U)$ is the continuation of $\Lambda .$ Since f is ${\mathcal {C}}$-topologically stable, f has the shadowing property on ${\mathcal {C}}.$ Thus if f has the $C^1$ robustly ${\mathcal {C}}$-topologically stable then it exactly is the notion of the $C^1$ robustly shadowing property on ${\mathcal {C}}.$ Thus as in the result of Sakai [15], ${\mathcal {C}}$ is hyperbolic. $\square$

We know that $\textrm{Diff}(M)$ is a Baire space in the $C^1$ topology. A residual subset of $\textrm{Diff}(M)$ is a countable intersection of open dense subsets. According to the Baire category theorem, a residual set is dense. We say that a property holds for the $C^1$ generic diffeomorphism f if it holds on a residual subset of $\textrm{Diff}(M).$

Theorem 3.3

There is a residual set ${\mathcal {G}}\subset \textrm{Diff}(M)$ such that for any $f\in {\mathcal {G}}$ and a chain transitive set ${\mathcal {C}}$ for f,  if f is ${\mathcal {C}}$-topologically stable then ${\mathcal {C}}$ is hyperbolic for f.

Proof

By Lemma 3.2, the diffeomorphism f is a $C^1$ generic f having the shadowing property on a locally maximal chain transitive set ${\mathcal {C}}$. From the result of Lee and Wen [7], we have ${\mathcal {C}}$ is hyperbolic for f. $\square$

For a chain transitive set ${\mathcal {C}}$, it is easily show that if a diffeomorphism f has the shadowing property on ${\mathcal {C}}$ then ${\mathcal {C}}$ is transitive. According to the above theorems, we have the following results.

Corollary 3.4

Let $\Lambda$ be a locally maximal transitive set of f. If a diffeomorphism f is $C^1$ robustly $\Lambda$-topologically stable then $\Lambda$ is hyperbolic.

Proof

By Theorem A, the transitive set $\Lambda$ satisfies a local star condition for f. By Lee [8], $\Lambda$ is hyperbolic for f. $\square$

According to Crovisier [3], a chain transitive set ${\mathcal {C}}$ is a transitive set. Thus by Theorem 3.3, we have the following.

Corollary 3.5

There is a residual set ${\mathcal {G}}\subset \textrm{Diff}(M)$ such that for any $f\in {\mathcal {G}}$ and $\Lambda$ is transitive set for f,  if f is $\Lambda$-topologically stable then $\Lambda$ is hyperbolic for f.