topological dynamics and ergodic theory. Recall that given a dynamical system $(X,T)$ and an open cover $\mathcal{U}$ of $X$ we define $$\mathcal{U}^n=\{U_0\cap T^{-1}(U_1)\cap \ldots \cap T^{-(n-1)}(U_{n-1}):U_j\in \mathcal{U}\text{ for }j=0,1,\ldots,n-1\}.$$ The *topological entropy* of $(X,T)$ with respect to $\mathcal{U}$ is given by $${h_\textrm{top}}(T,\mathcal{U})=\lim_{n\to\infty}\frac{1}{n}\log\mathcal{N}(\mathcal{U}^n),$$ where $\mathcal{N}(\mathcal{U}^n)$ denotes the smallest possible cardinality of an open cover of $X$ formed by elements of $\mathcal{U}^n$. We will denote by ${h_\textrm{top}}(T)$ the *topological entropy* of a map $T$ defined by $${h_\textrm{top}}(T)=\sup\{{h_\textrm{top}}(T,\mathcal{U}): \mathcal{U}\text{ is an open cover of }X\}.$$ For the proof of the existence of the limit above and basic properties of topological entropy see [@Walters chapter VII].

For the definition of the *measure-theoretic entropy* $h_{\mu}(T)$ of a $T$-invariant measure $\mu$ we refer for instance to [@Walters chapter IV].

Non-wandering set
-----------------

Given a dynamical system $(X,T)$ let $\Omega(T)$ be the *non-wandering set* of $T$, that is, $x\in X$ belongs to $\Omega(T)$ if for every neighborhood $U$ of $x$ there exists $n>0$ with $T^n(U)\cap U\neq\emptyset$. It is well known that $\Omega(T)$ is a closed invariant subset of $X$.

Chain recurrence
----------------

A $\delta$-chain (of length $m$) between $x$ and $y$ is any sequence ${\left\{x_n\right\}}_{n=0}^{m}$ such that $x=x_0$, $ y=x_m$, and $\rho(T(x_n),x_{n+1})<\delta$ for $0\le n <m$. A 