point $x$ is *chain recurrent* for $T$ if for every $\delta>0$ there is a $\delta$-chain from $x$ to $x$. The set of all chain recurrent points is denoted $\operatorname{CR}(T)$. Using compactness, we easily obtain that $\operatorname{CR}(T)$ is a closed set and for every $y\in \operatorname{CR}(T)$ there is $x\in \operatorname{CR}(T)$ such that $T(x)=y$, that is $T(\operatorname{CR}(T))=\operatorname{CR}(T)$.

A dynamical system $(X,T)$ is *chain recurrent* if $X=\operatorname{CR}(T)$. If for every $x,y\in X$ and every $\delta>0$ there exists a $\delta$-chain from $x$ to $y$ then $(X,T)$ is *chain transitive*.

Orbit segments and Bowen balls
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Let $a,b\in{{{\mathbb{N}}_0}}$, $a\leq b$. The *orbit segment* of $x\in X$ over $[a,b]$ is the sequence $$T^{[a,b]}(x)=(T^a(x),T^{a+1}(x),\ldots, T^b(x)).$$ We also write $T^{[a,b)}(x)=T^{[a,b-1]}(x)$. A *specification* is a family of orbit segments $$\xi=\{T^{[a_j,b_j]}(x_j)\}_{j=1}^n$$ such that $n\in{\mathbb{N}}$ and $b_j<a_{j+1}$ for all $1\le j <n$. The number of orbit segments in a specification is its *rank*.

The *Bowen distance between $x,y\in X$ along a finite set $\Lambda\subset{{{\mathbb{N}}_0}}$* is $$\rho^T_\Lambda(x, y) = \max\{\rho(T^j(x), T^j(y)) : j \in\Lambda\}.$$ By the *Bowen ball (of radius ${\varepsilon}$, centered at $x\in X$) along $\Lambda$* we mean the set $$B_\Lambda(x, {\varepsilon}) = \{y \in X : \rho^T_\Lambda(x, y) < {\varepsilon}\}.$$ If $\Lambda={\left\{0,1,\ldots,n-1\right\}}$ then we simply write $B_n(x, {\varepsilon}):=B_\Lambda(x, {\varepsilon})$ and 