C0-Stability for actions implies shadowing property

TL;DR

This paper constructs a specific class of stable actions on manifolds and proves that C0-stability guarantees the shadowing property in dimensions two or higher.

Contribution

It establishes a link between C0-stability and the shadowing property for actions on manifolds of dimension at least two, providing new insights into stability and shadowing.

Findings

C0-stable actions can be constructed explicitly.

C0-stability implies shadowing property in higher-dimensional manifolds.

Abstract

We will construct an action $\Phi$, C0 and C1-stable and we will prove that every C0-stable action acting in a manifold of dimensions greater or equal to two, have the shadowing property.

Equations

$$d(f(x_n),x_{n+1})<\delta ,\forall n\in Z.$$

$$d({\Phi }_s(x_g),x_{sg})<\delta ,\forall g\in G\text{mbox}and\forall s\in S.$$

$$d(x_g,{\Phi }_g(y))<\epsilon ,\forall g\in G.$$

$$d_S^0(\tilde{\Phi },\Phi ):=Ma{x}_{s\in S}\{d_{C^0}({\tilde{\Phi }}_s,{\Phi }_s)\},\text{mbox}for\tilde{\Phi },\Phi \in Act(G,X).$$

$$Ac{t}^1(G,M)=\{\Phi \in Act(G,M):{\Phi }_s\in C^1(M)\text{mbox}foralls\in S\}.$$

$$d_S^1(\tilde{\Phi },\Phi ):=Ma{x}_{s\in S}\{d_{C^1}({\tilde{\Phi }}_s,{\Phi }_s)\},\text{mbox}for\tilde{\Phi },\Phi \in Ac{t}^1(G,X).$$

$$||{\Phi }_s(x)-{\Phi }_s(y)||<1/2||x-y||\forall x,y\in I_{s^{{}^{\prime }}}\text{mbox}with{s}^{{}^{\prime }}\ne s^{-1}.$$

$$A_0=I_a\cup I_{a^{-1}}\cup I_b\cup I_{b^{-1}}\text{mbox}and$$

$$A_{n+1}=\left[{\Phi }_a(A_n)\cap {\Phi }_{a^{-1}}(A_n)\cap {\Phi }_b(A_n)\cap {\Phi }_{b^{-1}}(A_n)\right]\cap A_n.$$

$${\tilde{A}}_0={\tilde{I}}_a\cup {\tilde{I}}_{a^{-1}}\cup {\tilde{I}}_b\cup {\tilde{I}}_{b^{-1}},$$

$${\tilde{A}}_{n+1}=\left[{\tilde{\Phi }}_a({\tilde{A}}_n)\cap {\tilde{\Phi }}_{a^{-1}}({\tilde{A}}_n)\cap {\tilde{\Phi }}_b({\tilde{A}}_n)\cap {\tilde{\Phi }}_{b^{-1}}({\tilde{A}}_n)\right]\cap {\tilde{A}}_n$$

$$\text{mbox}andwecall{K}_{\tilde{\Phi }}^c=\cap {\tilde{A}}_n.$$

$$d({\tilde{\Phi }}_g(y_n),{\tilde{\Phi }}_g(z_n))>\alpha -2\epsilon >\frac{3\alpha }{4}\cdot$$

$$d_S^1(\tilde{\Phi },\Phi ):=Ma{x}_{{}_{s\in S}}\{d_{C^1}({\tilde{\Phi }}_s,{\Phi }_s),\text{mbox}for\tilde{\Phi },\Phi \in Ac{t}^1(F_2,{\mathcal{S}}^2)\}.$$

$$d({\Phi }_{b^{-1}}{\Phi }_a(z_n),z_{n+1})=d({\Phi }_{b^{-1}}{\Phi }_a(x_{(b^{-1}a{)}^n{g}_0}),x_{(b^{-1}a{)}^{n+1}{g}_0})\le$$

$$d({\Phi }_{b^{-1}}{\Phi }_a(x_{(b^{-1}a{)}^n{g}_0}),{\Phi }_{b^{-1}}(x_{a(b^{-1}a{)}^n{g}_0}))+d({\Phi }_{b^{-1}}(x_{a(b^{-1}a{)}^n{g}_0}),x_{(b^{-1}a{)}^{n+1}{g}_0})<{\delta }_1(\delta )+\delta .$$

$$\text{noindent}f({\Phi }_a({\tilde{x}}_{a^{-1}g}))=f({\Phi }_b({\tilde{x}}_{b^{-1}g}))={\tilde{x}}_g,$$

$$d_S^0(\tilde{\Phi },\Phi )<\rho .$$

$$\widetilde{\Phi}_{a}(\widetilde{x}_{g})=\widetilde{\Phi}_{a}(\widetilde{x}_{a^{-1}ag})\stackrel{{\scriptstyle\mbox{\scriptsize\begin{tabular}[]{c}$(\ref{pp})$\end{tabular}}}}{{=}}\widetilde{x}_{ag}.$$

$$d({\Phi }_g(h({\tilde{x}}_e)),x_g)=d(h{\tilde{\Phi }}_g({\tilde{x}}_e),x_g)=d(h({\tilde{x}}_g),x_g)$$

$$\le d(h({\tilde{x}}_g),{\tilde{x}}_g)+d({\tilde{x}}_g,x_g)<\epsilon /2+\eta <\epsilon .$$

Full text

Jorge Iglesias∗ and Aldo Portela∗

Universidad de La República. Facultad de Ingenieria. IMERL

Julio Herrera y Reissig 565. C.P. 11300

Montevideo, Uruguay

(Communicated by )

1 Introduction.

Given a topological group $G$ and a metric space $(X,d)$, a dynamical system is formally defined as a triplet $(G,X,\Phi)$, where $\Phi:G\times X\to X$ is a continuous function with $\Phi(g_{1},\Phi(g_{2},x))=\Phi(g_{1}g_{2},x)$ for all $g_{1},g_{2}\in G$ and for all $x\in X$. The map $\Phi$ is called an action of $G$ on $X$. It is possible to associate to each element of $g\in G$ a homeomorphism $\Phi_{g}:X\to X$ where $\Phi_{g}(x)=\Phi(g,x)$. For every $x\in X$ we define the orbit of $x$ as $O(x)=\{\Phi_{g}(x):\ g\in G\}$. A non-empty set $M\subset X$ is called minimal for the action if $\overline{O(x)}=M$ for any $x\in M$.

A group $G$ is finitely generated if there exists a finite set $S\subset G$ such that for any $g\in G$ there exist $s_{1},...,s_{n}\in S$ with $g=s_{1}.\cdots.s_{n}$. The set $S$ is called a finite generator of $G$. If $S$ is a finite generator of $G$ and for all $s\in S$ we have that $s^{-1}\in S$, then the set $S$ is called a finite symmetric generator.

For usual dynamical systems, that is when the group is $Z\!\!\!Z$ and the action is $\Phi(n,x)=f^{n}(x)$, we say that a sequence $\{x_{n}\}$ is $\delta$-pseudotrajectory if

[TABLE]

It is possible to generalize this concept to dynamical systems $(G,X,\Phi)$ as introduced in [OT]: A $G$-sequence in $X$ is a function $F:G\to X$. We denote this function by $\{x_{g}\}$ where $F(g)=x_{g}$. Let $S$ be a finite symmetric generator of $G$. For $\delta>0$ we say that a $G$-sequence $\{x_{g}\}$ is a $\delta$-pseudotrajectory if

[TABLE]

Given a dynamical system $(G,X,\Phi)$, we say that $\Phi$ has the shadowing property if for any $\varepsilon>0$ there exists $\delta>0$ such that for any $\delta$-pseudotrajectory $\{x_{g}\}$ there exists a point $y\in X$ with

[TABLE]

A finite symmetric generator $S$ of $G$ is uniformly continuous if for every $\varepsilon>$ 0 there exists $\delta>0$ such that $d(x,y)<\delta$ implies $d(\Phi_{s}(x),\Phi_{s}(y))<\varepsilon$ for every $s\in S$. It was proved in [OT, Proposition 1] that the shadowing property does not depend on the finite symmetric generator $S$ if it is uniformly continuous.

In the case of homeomorphism and diffeomorphisms (G=\mbox{Z!!!Z}) there is a strong relationship between stability and the shadowing property. An homeomorphism $f:X\to X$ is $C^{0}$-stable if for every $\varepsilon>0$, there exists $\delta>0$ such that if $g$ is another homeomorphism on $X$ with $d_{C^{0}}(f,g):=sup_{x\in X}\{d(f(x),g(x)\}<\delta$ then there exists a continuous and surjective function $h:X\to X$ with $hg=fh$ and $d_{C^{0}}(h,id)<\varepsilon.$ When $X$ is a manifold and $f:X\to X$ is a diffeomorphism, we say that $f$ is $C^{1}$-stable if there exists $\delta>0$ such that $g$ is another diffeomorphism with $d_{C^{1}}(f,g):=Max\{d_{C^{0}}(f,g),d_{C^{0}}(Df,Dg)\}<\delta$, then there exists a homeomorphisms $h:M\to M$ with $hg=fh$. In [W] was proved that every $C^{0}$-stable homeomorphism in a compact manifold of dimension greater or equal to two, has the shadowing property. This property is also valid for $C^{1}$-stable diffeomorphisms. The concepts of $C^{0}$ and $C^{1}$ stability can by generalized for actions as follows: let $S$ be a finite generating set for a group $G$. We denote by $Act(G,X)$ the set of actions in $X$, and we define a metric $d^{0}_{S}$ on $Act(G,X)$ by

[TABLE]

We say that an action $\Phi\in Act(G,X)$ is $C^{0}$- stable if for every $\varepsilon>0$, there exists $\delta>0$ such that if $\widetilde{\Phi}\in Act(G,X)$ with $d^{0}_{S}(\widetilde{\Phi},\Phi)<\delta$ then there exists a continuous and surjective function $h:X\to X$, with $h\widetilde{\Phi}_{g}=\Phi_{g}h,$ for every $g\in G$ and $d_{C^{0}}(h,id)<\varepsilon.$ It is not hard to prove that this definition does not depend on the generator $S$ if $X$ is a compact set (see [CL, Lemma 2.2]).

When $X$ is a manifold $M$, let $C^{1}(M)$ be the set of $C^{1}$ maps $f:M\to M$. Let $S$ be a finite generator of $G$. Let

[TABLE]

and consider

[TABLE]

An action $\Phi\in Act^{1}(G,M)$ is $C^{1}$- stable if there exists $\delta>0$ such that if $\widetilde{\Phi}\in Act^{1}(G,M)$ with $d^{1}_{S}(\widetilde{\Phi},\Phi)<\delta$, then there exists an homeomorphism $h:M\to M$ with $h\widetilde{\Phi}_{g}=\Phi_{g}h,$ for every $g\in G$.

We denote by $F_{2}$ the free group generated by two elements. Let us state our main result:

Theorem A

• Let $(F_{2},M,\Phi)$ be a dynamical system with $M$ a compact manifold of dimension greater or equal to two. If $\Phi$ is $C^{0}$-stable then $\Phi$ has the shadowing property.

An action $\Phi\in Act(G,X)$ is expansive if there exist $\alpha>0$ such that for every $x,y\in X$ with $x\neq y$ there exists $g\in G$ such that $d(\Phi_{g}(x),\Phi_{g}(y))>\alpha$. Let $X$ be a compact metric space. In [CL] was proved that if the action $\Phi\in Act(G,X)$ is expansive and has the shadowing property then $\Phi$ is $C^{0}$-stable. In section 2, we construct a $C^{0}$ and $C^{1}$-stable action of $G=F_{2}$ on $S^{1}$ that is not expansive. This shows that the expansive property is not a necessary condition for the stability.

2 Example of a $C^{0}$ and $C^{1}$-stable action.

In this section, we consider the free group $F_{2}$ with finite symmetric generator $S=\{a,a^{-1},b,b^{-1}\}$. We are going to construct a $C^{0}$ and $C^{1}$-stable action $\Phi$ in $\mathcal{S}^{2}$ whose minimal set $K$ is a Cantor set. The generator of the action will be $\Phi_{a}$ and $\Phi_{b}$ where $\Phi_{a},\Phi_{b}:\mathcal{S}^{2}\to\mathcal{S}^{2}$ will be defined later.

We begin defining a function $f_{a}:[0,+\infty]\to[0,+\infty]$, $f_{a}\in C^{1}$, as figure 1. Considerer the intervals $J_{a}$ y $J_{a^{-1}}=\overline{f^{-1}_{a}(J_{a})}^{c}$ such that (see figure 1):

$|f^{-1}_{a}(x)-f^{-1}_{a}(y)|>2|x-y|$ for all $x,y\in J_{a}$ (and consequently $|f_{a}(x)-f_{a}(y)|<1/2|x-y|$ for all $x,y\in J_{a}$ ). 2. 2.

$|f^{-1}_{a}(x)-f^{-1}_{a}(y)|<1/2|x-y|$ for all $x,y\in J_{a^{-1}}$.

Now we defining $\Phi_{a}:\mathcal{S}^{2}\to\mathcal{S}^{2}$ (in polar coordinates) and the intervals $I_{a}$ y $I_{a^{-1}}$ such that

• •

$\Phi_{a}(r,\theta)=(f_{a}(r),\theta).$

• •

$I_{a}=\{(r,\theta):\ \ r\in J_{a}\mbox{ and }\theta\in[0,2\pi)\}$ and $I_{a^{-1}}=\{(r,\theta):\ \ r\in J_{a^{-1}}\mbox{ and }\theta\in[0,2\pi)\}$

Note that $I_{a^{-1}}=\overline{\Phi^{-1}_{a}(I_{a})}^{c}$.

It is clear that $\Phi_{a}:\mathcal{S}^{2}\to\mathcal{S}^{2}$ is a north-south pole diffeomorphism, with $\Omega(\Phi_{a})=\{N_{a},S_{a}\}$, ( $S_{a}\in I_{a}$, $N_{a}\in I_{a^{-1}}$ ) and verifies

• •

$||\Phi^{-1}_{a}(x)-\Phi^{-1}_{a}(y)||>2||x-y||$ for all $x,y\in I_{a}$ (and consequently $||\Phi_{a}(x)-\Phi_{a}(y)||<1/2||x-y||$ for all $x,y\in I_{a}$ ).

• •

$||\Phi^{-1}_{a}(x)-\Phi^{-1}_{a}(y)||<1/2||x-y||$ for all $x,y\in I_{a^{-1}}$.

Now we consider $\Phi_{b}:\mathcal{S}^{2}\to\mathcal{S}^{2}$ north-south pole diffeomorphism, with $\Omega(\Phi_{b})=\{N_{b},S_{b}\}$ defined analogously to $\Phi_{a}$. Note that it is possible to construct $\Phi_{a}$ and $\Phi_{b}$ so that they verify the following properties:

$\overline{I}_{a}\cap\overline{I}_{s}=\emptyset$, for $s\in\{b,b^{-1}\}$ and $\overline{I}_{a^{-1}}\cap\overline{I}_{s}=\emptyset$, for $s\in\{b,b^{-1}\}$ 2. 2.

$||\Phi_{a}(x)-\Phi_{a}(y)||<1/2||x-y||$ for all $x,y\in I_{b}\cup I_{b^{-1}}$. 3. 3.

$||\Phi^{-1}_{a}(x)-\Phi^{-1}_{a}(y)||<1/2||x-y||$ for all $x,y\in I_{b}\cup I_{b^{-1}}$. 4. 4.

$||\Phi_{b}(x)-\Phi_{b}(y)||<1/2||x-y||$ for all $x,y\in I_{a}\cup I_{a^{-1}}$. 5. 5.

$||\Phi^{-1}_{b}(x)-\Phi^{-1}_{b}(y)||<1/2||x-y||$ for all $x,y\in I_{a}\cup I_{a^{-1}}$.

Let $\Phi$ be the action generated for $\Phi_{a}$ and $\Phi_{b}$. The figure 2 shows the dynamic of the action of $\Phi$ on the intervals $I_{a},I_{a^{-1}},I_{b}$ and $I_{b^{-1}}$.

The following properties are very useful for our purpose. Since they are not hard to prove we omit its proof.

Remark 1

If $s,s^{{}^{\prime}}\in\{a,a^{-1},b,b^{-1}\}$ then

[TABLE] 3. 2.

If $s\neq s{{}^{{}^{\prime}}{{}^{-1}}}$ then $\Phi_{s}(I_{s^{{}^{\prime}}})\subset I_{s}$. 4. 3.

$\Phi_{s}(I_{s})\cap I_{s^{-1}}=\emptyset$* for all $s\in\{a,b,a^{-1},b^{-1}\}$.*

Let $\{A_{n}\}$ be such that

[TABLE]

[TABLE]

Note that

• •

For any n\in\mbox{\mathcal{N}}, $A_{n}$ has $4.3^{n}$ connected components and $\overline{A_{n+1}}\subset int(A_{n})$.

• •

The diameter of the connected components fo $A_{n}$ goes to zero when $n$ goes to infinity.

• •

The Cantor set $K=\bigcap_{n\geq 1}A_{n}$ is a minimal set for the action $\Phi$ generated for $\Phi_{a}$ y $\Phi_{b}$ (see [N]).

Construction of the semiconjugacy.

For the action $\Phi$ we can think that the set $K$ is an ”attractor set” and that $A_{0}^{c}$ plays the role of a ”fundamental domain”. To prove that $\Phi$ is a $C^{0}$-stable action we will take $\delta>0$ such that if $d^{0}_{S}(\widetilde{\Phi},\Phi)<\delta$ then the dynamic of $\widetilde{\Phi}$ is given as figure 2. So we define $h:\widetilde{A}^{c}_{0}\to A^{c}_{0}$ close to identity, we extend it dynamically to $\mathcal{S}^{2}\setminus K$ and applying Lemma 2.5 it can be extended to $\mathcal{S}^{2}$.

Fix the discs $I_{a}$ and $I_{b}$. Let $\alpha=min\{d_{H}(\overline{I_{s}},\overline{I_{s^{{}^{\prime}}}}),\ s,s^{{}^{\prime}}\in\{a,a^{-1},b,b^{-1}\}\mbox{ and }s\neq s^{{}^{\prime}}\}$ be, where $d_{H}$ is the Hausdorff distance. Note that if $x,y\in K$, $x\neq y$, then there exists $g\in G$ such that $d(\Phi_{g}(x),\Phi_{g}(y))>\alpha$. For any other action $\widetilde{\Phi}$ considerer $\widetilde{I}_{a}=I_{a}$, $\widetilde{I}_{b}=I_{b}$, $\widetilde{I}_{a^{-1}}=\overline{\widetilde{\Phi}^{-1}_{a}(\widetilde{I}_{a})}^{c}$ and $\widetilde{I}_{b^{-1}}=\overline{\widetilde{\Phi}^{-1}_{b}(\widetilde{I}_{b})}^{c}$. Let

[TABLE]

[TABLE]

[TABLE]

Definition 2.1

Neighbourhood of stability.**

Given $\varepsilon>0$, $\varepsilon<\alpha/8$, let $\delta>0$ be, $\delta<\varepsilon$, such that if $\widetilde{\Phi}$ is an action with $d^{0}_{S}(\Phi,\widetilde{\Phi})<\delta$, then:

$\overline{\widetilde{I}_{s}}\cap\overline{\widetilde{I}_{s^{{}^{\prime}}}}=\emptyset$* for $s,s^{{}^{\prime}}\in\{a,a^{-1},b,b^{-1}\}$ with $s\neq s^{{}^{\prime}}$.* 2. 2.

If $s\neq s{{}^{{}^{\prime}}{{}^{-1}}}$ then $\overline{\widetilde{\Phi}_{s}(\widetilde{I}_{s^{{}^{\prime}}})}\subset\widetilde{I}_{s}$. 3. 3.

If $I$ is a connected component of $K_{\widetilde{\Phi}}$, then $diam(I)<\varepsilon/2.$ 4. 4.

For all $s\in\{a,b,a^{-1},b^{-1}\}$, $d(\widetilde{\Phi}^{-1}_{s}\Phi_{s}(x),x)<\varepsilon/2$ and $d(\Phi^{-1}_{s}\widetilde{\Phi}_{s}(x),x)<\varepsilon/2$ . 5. 5.

For all $x,y\in I_{s^{{}^{\prime}}}\cup\widetilde{I}_{s^{{}^{\prime}}}$, $s\neq(s^{{}^{\prime}})^{-1}$, $||\Phi_{s}(x)-\Phi_{s}(y)||<1/2||x-y||$ .

Note that if $\widetilde{\Phi}_{a}(r,\theta)=(\widetilde{f}_{a}(r),\theta)$ is a $C^{0}$-perturbed of $\Phi_{a}$, where $\widetilde{f}_{a}$ is as figure 3, then $K_{\widetilde{\Phi}}$ is not a Cantor set.

For the following lemmas, the action $\widetilde{\Phi}$ satisfies the five properties above.

The following remark will be very helpful.

Remark 2

$\widetilde{\Phi}_{s}(\widetilde{I}_{s})\cap\widetilde{I}_{s^{-1}}=\emptyset$* for all $s\in\{a,b,a^{-1},b^{-1}\}$.*

Some of our proofs are by induction in the length of the elements $g\in G$. Thus we need to define the length of an element $g\in G$. The elements of length one are $a,a^{-1},b$ and $b^{-1}$. The elements of length $n$ are obtained from the elements of length $n-1$ as follows: Let $g=s_{n-1}....s_{1}$ be an element of lengths $n-1$ with $s_{j}\in\{a,a^{-1},b,b^{-1}\}$. Then the elements of length $n$ generated by $g$ are $g^{{}^{\prime}}=s.g$ with $s\neq(s_{n-1})^{-1}$. The length of $g$ is denoted by $|g|$. It is clear that an element $g\in G$ can be written from $S$ in different ways, for example $g=gaa^{-1}$. Note that if $g=s_{{n}}....s_{{1}}$ with $s_{j}\in\{a,a^{-1},b,b^{-1}\}$, then $|g|\leq n$. We say that $g=s_{{n}}....s_{{1}}$ is written in its normal form if $|g|=n$. It is easy to prove that the normal representation is unique. From now on we will consider $g\in G$ written in its normal form.

Lemma 2.2

Let $\widetilde{J}$ be a connected component of $\widetilde{A}_{n}\setminus\widetilde{A}_{n+1}$ with $\widetilde{J}\subset\widetilde{I}_{s}$ and $g\in F_{2}$, $g=s_{j}...s_{1}$ with $s_{1}\neq s^{-1}$. Then $\widetilde{\Phi}_{g}(\widetilde{J})\subset\widetilde{A}_{n+j}\setminus\widetilde{A}_{n+1+j}$.

Proof:

As $\widetilde{J}\subset\widetilde{I}_{s}$ and $s_{1}\neq s^{-1}$, by item 2) in the election of $\delta$, $\widetilde{\Phi}_{s_{1}}(\widetilde{J})\subset\widetilde{I}_{s_{1}}.$ On the other hand, as $\widetilde{J}$ is a connected component of $\widetilde{A}_{n}\setminus\widetilde{A}_{n+1}$ then $\widetilde{\Phi}_{s_{1}}(\widetilde{J})$ is a connected component of $\widetilde{A}_{n+1}\setminus\widetilde{A}_{n+2}$. Again as $s_{2}\neq s_{1}^{-1}$, because $g$ is written in its normal form, then $\widetilde{\Phi}_{s_{2}}\widetilde{\Phi}_{s_{1}}(\widetilde{J})\subset\widetilde{I}_{s_{2}}$. As $\widetilde{\Phi}_{s_{1}}(\widetilde{J})$ is a connected component of $\widetilde{A}_{n+1}\setminus\widetilde{A}_{n+2}$ then $\widetilde{\Phi}_{s_{2}}\widetilde{\Phi}_{s_{1}}(\widetilde{J})$ is a connected component of $\widetilde{A}_{n+2}\setminus\widetilde{A}_{n+3}$. Reasoning inductively we obtain the thesis.

Remark 3

Given $\widetilde{J}$ a connected component of $\widetilde{A}_{n}\setminus\widetilde{A}_{n+1}$, there exists $g\in F_{2}$ such that $\widetilde{\Phi}_{g}(\widetilde{J})\subset\widetilde{A}_{0}^{c}$. 2. 2.

Given $\widetilde{J}$ a connected component of $\widetilde{A}_{n}\setminus\widetilde{A}_{n+1}$ with $\widetilde{J}\subset\widetilde{I}_{s}$. If $s^{{}^{\prime}}\neq s^{-1}$ then $\widetilde{\Phi}_{s^{{}^{\prime}}}(\widetilde{J})\subset\widetilde{A}_{n+1}$.

Lemma 2.3

Let $\widetilde{J}$ be a connected component of $\widetilde{A}_{n}\setminus\widetilde{A}_{n+1}$, with $n\geq 0$. Then there is a unique element $g_{{}_{\widetilde{J}}}$ of minimum length such that $\widetilde{\Phi}_{g_{{}_{\widetilde{J}}}}(\widetilde{J})\subset\widetilde{A}_{0}^{c}$.

Proof:

By Remark 3 item 1., there exists $g_{{}_{\widetilde{J}}}$ of minimum length such that $\widetilde{\Phi}_{g_{{}_{\widetilde{J}}}}(\widetilde{J})\subset\widetilde{A}_{0}^{c}$. Let $g_{{}_{\widetilde{J}}}=s_{r}...s_{1}$. Let $s$ be such that $\widetilde{J}\subset\widetilde{I}_{s}$. By Lemma 2.2 $s_{1}=s^{-1}$. If $n=0$ then $\widetilde{\Phi}_{g_{{}_{\widetilde{J}}}}(\widetilde{J})\subset\widetilde{A}_{0}^{c}$ and therefore $g_{{}_{\widetilde{J}}}=s^{-1}$.

If $n>0$ then $\widetilde{\Phi}_{s^{-1}}(\widetilde{J})$ is a connected component of $\widetilde{A}_{n-1}\setminus\widetilde{A}_{n}$. Reasoning as in the previous case, $s_{2}$ is determined. This proves the uniqueness of $g_{{}_{\widetilde{J}}}$.

Next, we will prove that for every action \widetilde{\Phi}\ ( that verifies the five conditions above) there exists a semiconjugacy $h:\mathcal{S}^{2}\to\mathcal{S}^{2}$ such that $h\widetilde{\Phi}_{g}=\Phi_{g}h$ for all $g\in F_{2}$.

Remark 4

As $\partial\widetilde{A}_{0}^{c}=\partial\widetilde{I}_{a}\cup\partial\widetilde{I}_{a^{-1}}\cup\partial\widetilde{I}_{b}\cup\partial\widetilde{I}_{b^{-1}}$, therefore

• •

If $\Phi_{g}(\widetilde{A}_{0}^{c})=\widetilde{A}_{0}^{c}$ then $g=e$.

• •

Let $x\in\partial\widetilde{A}_{0}^{c}$ be with $\Phi_{g}(x)\in\partial\widetilde{A}_{0}^{c}$, and $g\neq e$. If $x\in\partial\widetilde{I}_{s}$ then $g=s^{-1}$ and $\Phi_{g}(x)\in\partial\widetilde{I}_{s^{-1}}$.

We need that the following property to be satisfied. For this, we reduce $\delta$ as necessary so that in addition to complying with properties list in definiton 2.1, it verify the following additional property:

Given $\varepsilon>0$ (as in Definition 2.1), there exists $h:\overline{\widetilde{A}^{c}_{0}}\to\overline{A^{c}_{0}}$ homeomorphisms with $d(h(x),x)<\varepsilon$ and if $x\in\partial\widetilde{I}_{s}$ and $\widetilde{\Phi}_{s^{-1}}(x)\in\partial\widetilde{I}_{s^{-1}}$ then $h(x)=\Phi^{-1}_{s^{-1}}h\widetilde{\Phi}_{s}(x)$. This is possible by Remark 4.

By Lemma 2.3 we can extend $h$ dynamically to any connected component of $K^{c}_{\widetilde{\Phi}}$.

Lemma 2.4

For all $x\in K^{c}_{\widetilde{\Phi}}$, $d(h(x),x)<\varepsilon$.

Proof:

Let $x\in K^{c}_{\widetilde{\Phi}}$. If $x\in\widetilde{A}^{c}_{0}$ then, by definition of $h$, $d(h(x),x)<\varepsilon$.

If $x\in K^{c}_{\widetilde{\Phi}}\setminus\widetilde{A}^{c}_{0}$, then there exists n\in\mbox{\mathcal{N}} such that $x\in\widetilde{A}_{n}\setminus\widetilde{A}_{n+1}$. The proof will be done by induction in $n$. If $n=0$, $x$ belong $\widetilde{J}$, connected component of $\widetilde{A}_{0}\setminus\widetilde{A}_{1}$ with $\widetilde{J}\subset I_{s}$ for some $s\in\{a,a^{-1},b,b^{-1}\}$. Then $\widetilde{\Phi}_{s^{-1}}(\widetilde{J})\subset(\widetilde{A}_{0})^{c}$. As $h$ is dynamically defined, so $h(x)=\Phi_{s}h\widetilde{\Phi}_{s^{-1}}(x)$. As $\widetilde{\Phi}_{s^{-1}}(x)\in\widetilde{A}^{c}_{0}$, then $d(h\widetilde{\Phi}_{s^{-1}}(x),\widetilde{\Phi}_{s^{-1}}(x))<\varepsilon$.

By item 5 of definition 2.1, $d(\Phi_{s}h\widetilde{\Phi}_{s^{-1}}(x),\Phi_{s}\widetilde{\Phi}_{s^{-1}}(x))<\varepsilon/2$. By item 4 of definition 2.1, we have that $d(\Phi_{s}\widetilde{\Phi}_{s^{-1}}(x),x)<\varepsilon/2$ then $d(\Phi_{s}h\widetilde{\Phi}_{s^{-1}}(x),x)<\varepsilon$, therefore $d(h(x),x)<\varepsilon$.

Reasoning inductively, we finish the proof.

The following is a general lemma. As an application, $h$ is extended to $\mathcal{S}^{2}$.

Lemma 2.5

Extension lemma*.

Let $(X,d)$ be a compact metric space and $\Phi$, $\widetilde{\Phi}$ two actions of a group $G$. Let $X_{1}\subset X$ and $X_{2}\subset X$ be such that $\Phi_{g}(X_{1})=X_{1}$ and $\widetilde{\Phi}_{g}(X_{2})=X_{2}$ for all $g\in G$. There exists $\alpha>0$ such that:*

For all $x,y\in\overline{X_{1}}\setminus X_{1}$, $x\neq y$, there exists $g\in G$ such that $d(\Phi_{g}(x),\Phi_{g}(y))>\alpha$. 2. 2.

Given $\varepsilon>0$, $\varepsilon<\alpha/8$, there exists a homeomorphism $h:X_{2}\to X_{1}$, $h\widetilde{\Phi}_{g}=\Phi_{g}h$ for all $g\in G$, with $d(h(x),x)<\varepsilon$.

*Then, there exists a semiconjugacy $\overline{h}:\overline{X}_{2}\to\overline{X}_{1}$, $\overline{h}\widetilde{\Phi}_{g}=\Phi_{g}\overline{h}$ for all $g\in G$, with $d(\overline{h}(x),x)\leq\varepsilon$ and $\overline{h}|_{X_{2}}=h$.

Moreover, if for all $x,y\in\overline{X_{2}}\setminus X_{2}$, $x\neq y$, there exists $g\in G$ such that $d(\widetilde{\Phi}_{g}(x),\widetilde{\Phi}_{g}(y))>\alpha$, then $\overline{h}:\overline{X}_{2}\to\overline{X}_{1}$ is a homeomorphisms.

Proof:

Let $x\in\overline{X_{2}}\setminus X_{2}$ and $\{x_{n}\}$ with $x_{n}\to x$ and $x_{n}\in X_{2}$ for all n\in\mbox{\mathcal{N}}. Suppose that $\{h(x_{n})\}$ accumulate in points $y,z$ with $y\neq z$. Let $\{y_{n}\}$, $\{z_{n}\}$ be with $y_{n},z_{n}\to x$ and $h(y_{n})\to y$, $h(z_{n})\to z$. Note that $y,z\in\overline{X_{1}}\setminus X_{1}$ because $h$ is a homeomorphisms. As $y\neq z$ there exists $\Phi_{g}$ such that $d(\Phi_{g}(y),\Phi_{g}(z))>\alpha$. So, for $n$ big enough you have $d(\Phi_{g}(h(y_{n})),\Phi_{g}(h(z_{n})))>\alpha$. As $\Phi_{g}h(y_{n})=h\widetilde{\Phi}_{g}(y_{n})$, $\Phi_{g}h(z_{n})=h\widetilde{\Phi}_{g}(z_{n})$ then $d(h\widetilde{\Phi}_{g}(y_{n}),h\widetilde{\Phi}_{g}(z_{n}))>\alpha$. Since $d(h(x),x)<\varepsilon<\alpha/8$, we have that

[TABLE]

This is a contradiction because $y_{n},z_{n}\to x$.

So, we define $\overline{h}(x)=\lim h(x_{n})$. It is easy to prove that $\overline{h}$ is continuous, surjective, semiconjugacy ($\overline{h}\widetilde{\Phi}_{g}=\Phi_{g}\overline{h}$) and $d(\overline{h}(x),x)\leq\varepsilon$. This proves the first part of the lemma.

Let’s prove the second parte of lemma.

For this it is enough to prove that $\overline{h}$ is injective. Suppose that $\overline{h}(x)=\overline{h}(y)$ with $x\neq y$. Note that $x,y$ must belong to $\overline{X_{2}}\setminus X_{2}$.

As $x\neq y$ there exists $\widetilde{\Phi}_{g}$ such that $d(\widetilde{\Phi}_{g}(x),\widetilde{\Phi}_{g}(y))>\alpha$. As $d(\overline{h}(x),x)\leq\varepsilon$ then $d(\overline{h}\widetilde{\Phi}_{g}(x),\overline{h}\widetilde{\Phi}_{g}(y))\geq\frac{3\alpha}{4}$. Then $d(\Phi_{g}\overline{h}(x),\Phi_{g}\overline{h}(y))\geq\frac{3\alpha}{4}$. This is a contradiction because $\overline{h}(x)=\overline{h}(y)$. Therefore $\overline{h}$ es injective. So $\overline{h}$ is a homeomorphism.

Now, by the Lemma 2.5 we can extend $h$ to $\partial K^{c}_{\widetilde{\Phi}}$.

To finish, it is necessary to extend $h$ to $K_{\widetilde{\Phi}}$. Let $I$ be a connected component of $K_{\widetilde{\Phi}}$. Let $x,y\in\partial I\subset\partial K_{\widetilde{\Phi}}^{c}$. Note that the map $h$ is defined in the points $x,y$. We will prove that $h(x)=h(y)$. If $h(x)\neq h(y)$, as $h(x)$ and $h(y)$ are in $K$, then there exists $g\in G$ such that $d(\Phi_{g}(h(x)),\Phi_{g}(h(y)))>\alpha$. Therefore

$d(h\widetilde{\Phi}_{g}(x),h\widetilde{\Phi}_{g}(y))>\alpha$. As $d(h(x),x)\leq\varepsilon$ then $d(\widetilde{\Phi}_{g}(x),\widetilde{\Phi}_{g}(y))>3\alpha/4$. As $\widetilde{\Phi}_{g}(x)$ and $\widetilde{\Phi}_{g}(y)$ belong to the same connected component of $K_{\widetilde{\Phi}}$, by item 4. from definition of $\delta$, we obtain a contradition.

So $h$ is constant in $\partial I$ and we define $h(I)=h(x)$.

3 $C^{1}$-stability of action $\Phi$.

We considerer $Act^{1}(F_{2},\mathcal{S}^{2})$ the set of $C^{1}$ action with the distance

[TABLE]

Let $\Phi$ be as above. It is clear that it is possible to take $\delta>0$ such that $\widetilde{\Phi}$ is a $C^{1}$ action with $d^{1}_{S}(\widetilde{\Phi},\Phi)<\delta$ and:

The action $\widetilde{\Phi}$ verifies the properties of definition 2.1. 2. 2.

The maps $\widetilde{\Phi}_{a}$ and $\widetilde{\Phi}_{b}$ are north-south pole $C^{1}$ diffeomorphisms, $K_{\widetilde{\Phi}}$ is a Cantor set and for any $x,y\in K_{\widetilde{\Phi}}$, $x\neq y$, there exists $g\in F_{2}$ such that $d(\widetilde{\Phi}_{g}(x),\widetilde{\Phi}_{g}(y))>\alpha$.

By Lemma 2.5, the action $\Phi$ is $C^{1}$-stable.

4 $C^{0}$-stability implies shadowing property.

Here we will considerer the dynamical system $(F_{2},M,\Phi)$ where $M$ is a compact manifold of dimension greater or equal two.

Recall that given $g\in F_{2}$ we denoted by $|g|$ the length of $g$.

Given $\delta>0$ and n\in\mbox{\mathcal{N}}, we say that $\{x_{g}\}_{|g|\leq n}$, $x_{g}\in M$, is a $\delta-n$ pseudotrajectory if $d(x_{sg},\Phi_{s}(x_{g}))<\delta$ for all $s\in\{a,a^{-1},b,b^{-1}\}$, for all $g\in F_{2}$ with $|g|\leq n$.

Lemma 4.1

If for all $\varepsilon>0$ there exists $\delta>0$ such that if for all n\in\mbox{\mathcal{N}} and for all $\{x_{g}\}_{|g|\leq n}$ $\delta-n$, pseudotrajectory there exists $y\in M$ such that $d(\Phi_{g}(y),x_{g})<\varepsilon$, $|g|\leq n$, then $\Phi$ has the shadowing property.

Proof:

Given $\varepsilon$, choose $\delta$ as in the statement of the lemma. Let $\{x_{g}\}$ be a $\delta$ pseudotrajectory. For any n\in\mbox{\mathcal{N}} we consider $\{x_{g}\}_{|g|\leq n}$ , that is a $\delta-n$ pseudotrajectory included in $\{x_{g}\}$. By hypotheses, there exist $y_{n}$ such that $d(\Phi_{g}(y_{n}),x_{g})<\varepsilon$ for $|g|\leq n$.

Let $\{y_{n_{k}}\}$ be a subsequence of $\{y_{n}\}$ such that $y_{n_{k}}\to y$. Given $g\in F_{2}$ let $n_{k}$ be such that $|g|\leq n_{k}$ and $d(\Phi_{g}(y),\Phi_{g}(y_{n_{k}}))<\varepsilon.$

Then $d(\Phi_{g}(y),x_{g})\leq d(\Phi_{g}(y),\Phi_{g}(y_{n_{k}}))+d(\Phi_{g}(y_{n_{k}}),x_{g})<2\varepsilon.$ And the proof is finished.

For the proof of the next lemma see [NS, Lemma 13].

Lemma 4.2

Let M be a compact manifold of dimension greater than or equal to two. Suppose a finite collection $\{(p_{i},q_{i})\in M\times M\mbox{for }i=1,...,r\}$ is specified together with a small $\lambda>0$ such that

$d(p_{i},q_{i})<\lambda$* for all $i$, and* 2. 2.

if $i\neq j$ then $p_{i}\neq p_{j}$ and $q_{i}\neq q_{j}$.

Then there exists a diffeomorphism $f:M\to M$ such that

(a) $d(f,id)<2\pi\lambda$, and

(b) $f(p_{i})=q_{i}$ ($1\leq i\leq r$).

Lemma 4.3

If the action $\Phi$ is $C^{0}$-stable then the map $\Phi_{a^{-1}}\Phi_{b}$ is $C^{0}$-stable.

Proof:

Let $f$ be $C^{0}$-close to $\Phi_{a^{-1}}\Phi_{b}$. So $\Phi_{a}f$ is $C^{0}$-close to $\Phi_{b}$. We considerer the action $\widetilde{\Phi}$ generated by $\Phi_{a}$ and $\Phi_{a}f$. As $\Phi$ is $C^{0}$-stable, there exists a semiconjugacy $h$ such that : $h\Phi_{a}=\Phi_{a}h$ and $h\Phi_{a}f=\Phi_{b}h$. Therefore $\Phi_{a}hf=\Phi_{b}h$ and $hf=\Phi_{a^{-1}}\Phi_{b}h$.

Remark 5

As the map $\Phi_{a^{-1}}\Phi_{b}$ is $C^{0}$-stable there is no an open set $U\subset M$ such that every point of $U$ is a periodic point, with all of them with the same period.

Lemma 4.4

Let $\Phi$ be a $C^{0}$-stable action. Given $\eta>0$ there exists $\delta_{0}>0$ such that if $\{x_{g}\}$ is a $\delta$-pseudotrajectory for $\Phi$ with $\delta<\delta_{0}$, then there exists $\{\widetilde{x}_{g}\}$ such that:

• •

$d(x_{g},\widetilde{x}_{g})<\eta$* and*

• •

$\Phi_{a}(\widetilde{x}_{a^{-1}g})=\Phi_{b}(\widetilde{x}_{b^{-1}g})$* for all $g\in F_{2}.$*

Proof:

Let $\{x_{g}\}$ be a $\delta$-pseudotrajectory. We will define an equivalence relation in $\{x_{g}\}$. We say that $x_{g}$ is equivalent to $x_{g^{\prime}}$ if there exists n\in\mbox{Z!!!Z} such that $g=(b^{-1}a)^{n}g^{\prime}$. Let $[x_{g}]$ be the class of element of $x_{g}$. Note that there exists $g_{0}\in F_{2}$ with minimum length such that $[x_{g}]=[x_{g_{0}}]=\{x_{(b^{-1}a)^{n}g_{0}}:n\in Z\}$. Therefore the class $[x_{g}]$ can be thought as a sequence $\{z_{n}\}$ such that $z_{n}=x_{(b^{-1}a)^{n}g_{0}}$, n\in\mbox{Z!!!Z}.

Let’s prove that $\{z_{n}\}$ is a $\delta+\delta_{1}$-pseudotrajectory for the map $\Phi_{b^{-1}}\Phi_{a}$, where $\delta_{1}(\delta)=Max\{d(\Phi_{b^{-1}}(x),\Phi_{b^{-1}}(y)):\ \ d(x,y)<\delta\}$.

[TABLE]

[TABLE]

As the map $\Phi_{b^{-1}}\Phi_{a}$ is $C^{0}$-stable, then it has the shadowing property (see [W] ). So given $\eta>0$ there exists $\delta^{{}^{\prime}}_{0}$ such that if $\delta_{1}(\delta)+\delta<\delta^{{}^{\prime}}_{0}$ then there exists $y$ such that

$d((\Phi_{b^{-1}}\Phi_{a})^{n}(y),z_{n})<\eta$.

Therefore, for each $[x_{g_{0}}]$ there exists $y_{g_{0}}\in M$ such that

$d((\Phi_{b^{-1}}\Phi_{a})^{n}(y_{g_{0}}),x_{(b^{-1}a)^{n}g_{0}})<\eta$.

Now, we define $\{\widetilde{x}_{g}\}$ such that $\widetilde{x}_{g}=(\Phi_{b^{-1}}\Phi_{a})^{n}(y_{g_{0}})$ if $g=(b^{-1}a)^{n}g_{0}$.

Note that $x_{a^{-1}g}$ is equivalent to $x_{b^{-1}g}$ , so $\Phi_{b^{-1}a}(\widetilde{x}_{a^{-1}g})=\widetilde{x}_{b^{-1}g}$. Therefore $\Phi_{a}(\widetilde{x}_{a^{-1}g})=\Phi_{b}(\widetilde{x}_{b^{-1}g})$ and $d(x_{g},\widetilde{x}_{g})<\eta$. So given $\eta>0$ it is enough to take $\delta_{0}$ such that $\delta_{1}(\delta_{0})+\delta_{0}<\delta^{{}^{\prime}}_{0}$.

Proof of Theorem A.

Given $\eta>0$, let $\delta_{0}$ be given by Lemma 4.4. Let $\{x_{g}\}$ be a $\delta$ pseudotrajectory with $\delta<\delta_{0}$. For each n\in\mbox{\mathcal{N}}, let $\{x_{g}\}_{|g|\leq n}$ be, a $\delta-n$ pseudotrajectory. Again, by Lemma 4.4 there exists $\{\widetilde{x}_{g}\}$ such that $d(x_{g},\widetilde{x}_{g})<\eta$ for $|g|\leq n$. By construction, $\widetilde{x}_{g}=(\Phi_{b^{-1}}\Phi_{a})^{m}(y_{g_{0}})$. As $|g|\leq n$, by Remark 5, it is possible to take $y_{g_{0}}$ such that $\widetilde{x}_{g}\neq\widetilde{x}_{g^{{}^{\prime}}}$ if $g\neq g^{{}^{\prime}}$ with $|g|\leq n$ and $|g^{{}^{\prime}}|\leq n$.

By Lemma 4.1 it is enough to prove the Theorem A for a $\delta-n$ pseudotrajectory.

Given $\varepsilon>0$ let

• •

$\rho>0$, $\rho<\varepsilon/2$, such that, if $\widetilde{\Phi}$ is an action with $d^{0}_{S}(\widetilde{\Phi},\Phi)<\rho$ then there exists $h:M\to M$, $d(h,id)<\varepsilon/2$ and $h\widetilde{\Phi}_{g}=\Phi_{g}h$ for all $g\in F_{2}$.

• •

$\eta>0$, $\eta<\varepsilon/2$. We take $\delta<\delta_{0}$ small enough such that for any $\{\widetilde{x}_{g}\}$ as above, ($|g|\leq n$) with $d(x_{g},\widetilde{x}_{g})<\eta$ there exists a diffeomorphisms $f:M\to M$, given by Lemma 4.2, such that:

[TABLE] 2. 2.

If $\widetilde{\Phi}$ is the action generated by $\widetilde{\Phi}_{a}=f\Phi_{a}$ and $\widetilde{\Phi}_{b}=f\Phi_{b}$, then

[TABLE]

Let’s prove that $\widetilde{\Phi}_{s}(\widetilde{x}_{g})=\widetilde{x}_{sg}$ for all $s\in\{a,a^{-1},b,b^{-1}\}$, for all $g\in F_{2}$. It is enough to prove that $\widetilde{\Phi}_{s}(\widetilde{x}_{g})=\widetilde{x}_{sg}$ for all $s\in\{a,b\}$, for all $g\in F_{2}$. Suppose $s=a$. If $s=b$ the proof is analogous.

[TABLE]

As $d^{0}_{S}(\widetilde{\Phi},\Phi)<\rho$, then there exists $h:M\to M$ such that $h\widetilde{\Phi}_{g}=\Phi_{g}h$ for all $g\in F_{2}$, with $d(h(x),x)<\varepsilon/2$ . So

[TABLE]

[TABLE]