Countably Expansiveness for Continuous Dynamical Systems

Let X be a compact metric space with a metric d and $f:X\to X$ be a homeomorphism. Utz [1] introduced a dynamic property, which is called expansiveness. It means that, if two orbits stay within a small distance, then the orbits are the same. That is, a homeomorphism f is expansive if there is an expansive constant $e>0$ such that for any $x\ne y\in X$ there is $i\in \mathbb{Z}$ satisfying $d(f^i(x),f^i(y))>e.$ From the definition of the expansiveness, it is possible to consider the set We can easily check that f is expansive if and only if ${\mathsf{\Phi }}_{\delta }^f(x)=\{x\}$ for all $x\in X.$

Now, we have a natural question:

It is easy to see that if f is expansive then f is N-expansive. Now, we introduce another notion of expansiveness, which is a general notion of expansiveness.

Note that the relationship with among those notions is

On the other hand, from the stochastic point of view, Morales and Sirvent [2] introduced a general notion of expansiveness by using a measure. For the Borel $\sigma$-algebra $\beta$ on X, we denote $\mathcal{M}(X)$ the set of Borel probability measures on X endowed with the weak${}^{\ast }$ topology. Let ${\mathcal{M}}^{\ast }(X)$ be the set of nonatomic measure $\mu \in \mathcal{M}(X).$

In among the notions, a remarkable notion is measure expansiveness (which was introduced by Morales [2]). It is exactly same as countably expansiveness (see [3]). That is, Artigue and Carrasco-Olivera [3] considered a relationship between the measure expansiveness and the countably expansiveness.

Let M be a compact connected smooth manifold, and let Diff${}^1(M)$ 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 $\parallel ·\parallel$ on the tangent bundle $TM.$ In dynamical systems, the concept of expansiveness [1] is a useful notion for studying stability theory. In fact, Mañé [4] showed that if a diffeomorphism f of M is $C^1$ stably expansive then it is quasi-Anosov. Here, we say that f is quasi-Anosov if, for all $v\in TM\backslash \{\mathbf{0}\},$ the set $\{\parallel D{f}^n(v)\parallel :n\in \mathbb{Z}\}$ is unbounded.

Later, many mathematicians studied stability theory using the various types of expansiveness [5,6,7,8]. For instance, Moriaysu, Sakai, and Yamamoto [8] showed that if a diffeomorphism f of M is $C^1$ stably measure expansive then it is quasi-Anosov. The result is a generalization of the result of Mañé [4].

On the other hand, it is very important to extend from diffeomorphisms to vector fields (flows). In fact, many researchers studied the various aspects of flows, such as thermodynamics-Hamiltonian systems [9], nonlinear systems [10], and chaos systems [11,12].

From the result of [4], Moriyasu, Sakai, and Sun [7] extended expansive diffeomorphisms to vector fields about the $C^1$ stably point of view. That is, they showed that if a vector field X is $C^1$ stably expansive then it is quasi-Anosov. Lee and Oh [5] showed that if a vector field X is $C^1$ stably measure expansive then it is quasi-Anosov (We refer to the basic definitions related to the vector fields below.).

In this article, we introduce another type of countably expansive vector fields which is different than weak expansiveness in [6]. In addition, we establish some examples of the countably expansiveness for homeomorphisms and flows, such as shift map and suspension flow by applying the rotation map on the circle. Moreover, we prove that if a vector field X of a compact connected manifold M is $C^1$ stably countably expansive, then it is quasi-Anosov which is a general result of Moriyasu, Sakai, and Sun [7]. Furthermore, we have that if a vector field X of a compact connected manifold M is $C^1$ stably expansive, weak expansive, and countably expansive then it is quasi-Anosov.

In this paper, we focus on countably expansiveness which is defined as the following remark.

Symbolic systems can be used to “code” some smooth systems. Indeed, to study of symbolic dynamics is the research of a specific class of the shift transformation in a sequence space. In addition, it provides more motivation of the relationships between topological and smooth dynamics. The properties of symbolic dynamical systems give a rich source of examples and counterexamples for topological dynamics and ergodic theory.

The set of all infinite sequences of 0s and 1s is called the sequence space of 0 and 1 or the symbol space of 0 and 1 is denoted by ${\Sigma }_2.$ More precisely, ${\Sigma }_2=\{(s_0{s}_1{s}_2\cdots )|s_i=0\mathrm{or}{s}_i=1(\forall i)\}.$ We often refer to elements of ${\Sigma }_2$ as points in ${\Sigma }_2.$ Shift map $\sigma :{\Sigma }_2\to {\Sigma }_2$ is defined by In short, the shift map “deletets” is the first coordinate of the sequence, for example $\sigma (01110101\cdots )=1110101\cdots$ (for more details, see [13]).

Since $|s_i-t_i|$ is either 0 or 1, we know that

The shift map is continuous; it is clear that two points are close if and only if their initial coordinates are same. The more the coordinates are the same before they are different, the closer they are to each other. Then, we know that the set of periodic points of the shift map is dense in ${\Sigma }_2$ (the shift map has $2^n$ periodic points of period n). We put the set then $\overline{D}={\Sigma }_2$ where $\overline{D}$ is a dense set of $D.$ In fact, we can check the above facts by the following example.

On the other hand, we need to check some properties of countably expansive homeomorphism, which are used to prove the lemmas and theorems below.

Lee, Morales, and Thach [6] characterized the countably expansive flows in measure-theoretical terms, which is extended the result of [3] in the discrete case, called weak expansive flows. They showed that a flow is countably expansive if and only if the flow is weak measure expansive.

Here, $\mathcal{C}$ denotes the set of continuous maps $c:\mathbb{R}\to \mathbb{R}$ with $c(0)=0$ and ${\varphi }_{\mathbb{R}}(x)=\{{\varphi }_t(x):t\in \mathbb{R}\}.$

Now, we introduce the new notion of countably expansive flows by using a dense subset D of X and consider the examples showing the countably expansive property, very well. A continuous flow of X such that

Denote by

Now, let $\tau :{\Sigma }_2\to \mathbb{R}$ be a continuous function and consider the space with $(x,\tau (x))\sim (\sigma (x),0)$ for each $x\in {\Sigma }_2.$ The suspension flow over $\sigma$ with height function $\tau$ is the flow ${({\varphi }_t^{\tau ,f})}_{t\in \mathbb{R}}$ on $Y^{\tau ,f}$ defined by

More precisely, for all $x\in {\Sigma }_2,$ ${\sigma }^0(x)=0$ and ${\sigma }^{n+1}(x)={\sigma }^n(x)+\sigma (\tau (x))$ for all $n\in \mathbb{Z}.$ For all $(x,s)\in Y^{\tau ,f}$ and $t\in \mathbb{R},$ there is a unique $n\in \mathbb{Z}$ such that ${\sigma }^n(x)\le s+t<{\sigma }^{n+1}(x);$ we set

The Bowen–Walters distance (Definition 2 of [14]) makes $Y^{\tau ,f}$ a compact metric space where a neighborhood of a point $(x,s)\in Y$ contains all the points of ${\varphi }_t^{\tau ,f}(w,s)\in Y^{\tau ,f}$ where $\left|t\right|$ is small and w is close to $x.$ With respect to the topology generated, ${\varphi }_t^{\tau ,f}$ is a homeomorphism on $Y^{\tau ,f}$ for all $t\in \mathbb{R}.$

Carrasco-Olivera and Morales [15] extended the concept of expansive measure from homeomorphism [2] to flows. They (respectively [6]) showed that a homeomorphism of a compact metric space is measure expansive (respectively, countable expansive) if its suspension flow is. The following theorem says that the case of countably expansiveness, which is defined in this paper, is also satisfied.

By the following examples, it is easy to see that a suspension flow over an irrational rotation map on the unit circle $S^1$ is countably expansive by applying the Theorem 1.

In addition, we can see that the following example satisfies Theorem 1.

On the other hand, we can find a dense subset of ${\Sigma }_2$ as following remark.

Moreover, we can show that Theorem 1 holds by using the above remark for the case of the suspension flow over a shift map as following example.

Recall that M is a compact connected smooth manifold, d is the distance on M induced from a Riemannian metric $\parallel ·\parallel$ on the tangent bundle $TM.$ Denote by ${\mathfrak{X}}^1(M)$ the set of all $C^1$ vector fields of M endowed with the $C^1$ topology. Then, every $X\in {\mathfrak{X}}^1(M)$ generates a $C^1$ flow $X_t:M\times \mathbb{R}\to M$ satisfying $X_s\circ X_t=X_{s+t}$ for all $s,t\in \mathbb{R}$, $X_0=1_d$ and $d{X}_t{(x)/dt|}_{t=0}=X(x)$ for any $x\in M$. Here, $X_t$ is called the integrated flow of $X.$ Throughout this paper, for $X,Y,\dots \in {\mathfrak{X}}^1(M)$, denote the integrated flows by $X_t,Y_t,\dots$, respectively.

For $x\in M$, denote by ${\mathcal{O}}_X(x)$ the orbit $\{X_t(x):t\in \mathbb{R}\}$ of the flow $X_t$ (or X) through x. A point $x\in M$ is singular of X if $X(x)={\mathbf{0}}_x,$ $\mathrm{Sing}(X)$ denotes the set of singular points of $X.$ It is said that a point p is periodic if $X_{\pi (p)}(p)=p$ for some $\pi (p)>0,$ but $X_t(p)\ne p$ for all $0<t<\pi (p),$ $PO(X_t)$ denotes the set of periodic points of $X.$ A point p is regular if $x\notin PO(X_t)$ and $x\notin \mathrm{Sing}(X).$ The set of non-wandering points of X, denoted by $\mathsf{\Omega }(X_t)$ then we can see that

A flow $X_t$ of M is expansive if for given $ϵ>0$ there is a expansive constant $\delta >0$ such that if $x,y\in X$ satisfying $d(X_t(x),X_{h(t)}(y))\le \delta$ for some $h\in \mathcal{H}$ and all $t\in \mathbb{R}$ then $y\in X_{[-ϵ,ϵ]}(x)$, where $\mathcal{H}$ denotes the set of increasing continuous maps $h:\mathbb{R}\to \mathbb{R}$ fixing $0.$

First, we can check some properties of countably expansive flows as following lemmas.

We can see that the singular points of countably expansive flows are isolated by the below lemma.

A closed $X_t$-invariant subset $\mathsf{\Lambda }$ is hyperbolic if there exist constants $C>0$, $\lambda >0$ and a splitting $T_{x}M=E_x^s\oplus \langle X(x)\rangle \oplus E_x^u(x\in \mathsf{\Lambda })$ satisfying the tangent flow $D{X}_t:TM\to TM$ has invariant continuous splitting and for $x\in \mathsf{\Lambda }$ and $t>0.$ We say that $X\in {\mathfrak{X}}^1(M)$ is Anosov when M is hyperbolic for $X_t.$

We say that a vector field X is $AxiomA$ if $PO(X_t)$ is dense in $\mathsf{\Omega }(X_t)\backslash \mathrm{Sing}(X)$ and $\mathsf{\Omega }(X_t)$ is hyperbolic. For Axiom A vector field $X,$ we know that $\mathsf{\Omega }(X_t)$ is equal to the union of each basic set ${\mathsf{\Lambda }}_i(1\le i\le l)$ of $X.$ Note that the basic set is closed, invariant, and transitive. A collection of basic sets ${\mathsf{\Lambda }}_{i_1},\cdots ,{\mathsf{\Lambda }}_{i_k}$ of X is called a cycle if, for each $j=1,2,\cdots ,k,$ there exists $a_j\in \mathsf{\Omega }(X_t)$ such that $\alpha (a_j)\subset {\mathsf{\Lambda }}_{i_j}$ and $\omega (a_j)\subset {\mathsf{\Lambda }}_{i_{j+1}}(k+1\equiv 1).$ We say that a vector field X has no cycle if there exist no cycles among the basic sets of $X.$

For any hyperbolic periodic point x of X, the sets are the stable manifold and unstable manifold of x, respectively. For Axiom A vector field $X\in {\mathfrak{X}}^1(M),$ we say that X has the quasi-transversality condition if $T_x{W}^s(x)\cap T_x{W}^u(x)=\{{\mathbf{0}}_{\mathbf{x}}\}$ for any $x\in M$.

The exponential map defined by ${\exp }_x:T_{x}M(1)\to M$ for all $x\in M$ where $T_{x}M(r)=\{v\in T_{x}M:\parallel v\parallel \le r\}.$ Let $M_X=\{x\in M:X(x)\ne {\mathbf{0}}_x\}.$ For any $x\in M_X,$ we set where $N_x(r)=N_x\cap T_{x}M(r)$ for $0<r\le 1.$

Let $\mathcal{N}={\bigcup }_{x\in M_X}{N}_x$ be the normal bundle on $M_X.$ Then, we present a linear Poincaré flow for X on $\mathcal{N}$ by where ${\pi }_{N_x}:T_{x}M\to N_x$ is the natural projection along the direction of $X(x),$ and $D_x{X}_t$ is the derivative map of $X_t.$ We say that $X\in {\mathfrak{X}}^1(M)$ is quasi-Anosov if ${\sup }_{\mathrm{t}\in \mathbb{R}}\left|\right|{\mathsf{\Psi }}_{\mathrm{t}}(\mathit{v})\left|\right|<\infty$ for $v\in \mathcal{N}$ then $v=\mathbf{0}$.

The main theorem of this paper stated as follows.

Now, let us prove the above theorem. To show this, we first need following lemma.

Denote ${\mathfrak{X}}^{\ast }(M)$ as the set of $X\in {\mathfrak{X}}^1(M)$ with the property that there is a $C^1$ neighborhood $\mathcal{U}(X)$ of X such that every $\gamma \in$ PO$(Y_t)$ is hyperbolic for $Y\in \mathcal{U}(X).$ It was proved by [17] that $X\in {\mathfrak{X}}^{\ast }(M)$ if and only if X satisfies Axiom A without cycle condition.

For given $p\in M_X,t\in \mathbb{R},$ take a constant $r>0,$ a $C^1$ map $\tau :{\mathsf{\Pi }}_{p,r}\to \mathbb{R}$ such that $\tau (p)=t$ and $X_{\tau (y)}(y)\in {\mathsf{\Pi }}_{X_t(p),1}$ for $\forall y\in {\mathsf{\Pi }}_{p,r}.$ The Poincaré map $f_{p,t}:{\mathsf{\Pi }}_{p,r_0}\to {\mathsf{\Pi }}_{X_t(p),1}$ is given by For given $ϵ>0,$ we denote by ${\mathcal{N}}_{ϵ}({\mathsf{\Pi }}_{p,r})$ the set of diffeomorphisms $\xi :{\mathsf{\Pi }}_{p,r}\to {\mathsf{\Pi }}_{p,r}$ satisfying supp$(\xi )\subset {\mathsf{\Pi }}_{p,\frac{r}{2}}$ and $d_{C^1}(\xi ,{\mathtt{I}}_{\mathtt{d}})<ϵ.$ Here, $d_{C^1}$ is the $C^1$ metric, ${\mathtt{I}}_{\mathtt{d}}:{\mathsf{\Pi }}_{p,r}\to {\mathsf{\Pi }}_{p,r}$ is the identity map and supp$(\xi )=\overline{{\mathsf{\Pi }}_{p,r}}$ where it differs from ${\mathtt{I}}_{\mathtt{d}}.$