$\rho^T_n(x, y)=\rho^T_\Lambda(x, y)$.

Natural extension
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The *inverse limit space* of a surjective dynamical system is the space $$X_T=\{(x_1,x_2,x_3,\ldots)\in X^\infty:T(x_{i+1})=x_i \text { for all }i\in{\mathbb{N}}\}.$$ We equip $X_T$ with the subspace topology induced by the product topology on $X^\infty$. The map $T$ is called a *bonding map*. The map $\sigma_T\colon X_T\to X_T$ given by $$\sigma_T(x_0,x_1,x_2,\ldots)=(T(x_0),T(x_1),T(x_2),\ldots)=(T(x_0),x_0,x_1,\ldots).$$ is called the *shift homeomorphism* and the invertible dynamical system $(X_T,\sigma_T)$ is a *natural extension* of $(X,T)$. Note that if $(X,T)$ is invertible then $(X,T)$ and $(X_T,\sigma_T)$ are conjugate. If $T$ is not invertible, then $(X,T)$ is only a factor of $(X_T,\sigma_T)$.

Dynamical systems $(X,T)$ and $(X_T,\sigma_T)$ share many dynamical properties. For example, it is not hard to check that one of them is transitive, mixing or has a specification(-like) property if and only if the other has the respective property. It was proved in [@ChenLi] that the same equivalence holds for the shadowing property. Furthermore, the invariant measures of $(X,T)$ and $(X_T,\sigma_T)$ can be identified by a natural entropy preserving bijection. Hence, ${h_\textrm{top}}(T)={h_\textrm{top}}(\sigma_T)$ (see [@Ye95] for a more general statement).

Expansiveness
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An (invertible) dynamical system $(X,T)$ is *positively expansive (expansive)* if there is a constant $c>0$ such that if $x,y\in X$ satisfy $d(T^n(x),T^n(y))<c$ 