for all $n\in {{{\mathbb{N}}_0}}$ (all $n\in {\mathbb{Z}}$), then $x=y$.

Two-sided shift spaces and Axiom A diffeomorphisms are expansive (see [@DGS]). One-sided shift spaces are positively expansive. If $(X,T)$ is invertible and positively expansive, then $X$ is a finite set (see [@Richeson]). If a dynamical system is expansive or positively expansive, then its natural extension is expansive, but the converse is not true (see [@AH], Theorem 2.2.32(3)).

Specification property
======================

The periodic specification property was introduced by Bowen [@Bowen71] as a consequence of topological mixing of an axiom A diffeomorphism. Roughly speaking, the specification property allows to approximate segments of orbits by a single orbit, provided that these segments are sparse enough in time. Recall that a diffeomorphism $T\colon M\to M$ of a smooth compact manifold satisfies *Smale’s Axiom A* if the periodic points of $T$ are dense in the non-wandering set $\Omega(T)$ and the tangent bundle of $M$ restricted to $\Omega(T)$, denoted $T\Omega(M)$, has a continuous splitting $T\Omega(M)=E_s\oplus E_u$ into subspaces invariant under the derivative $DT$ such that the restrictions $DT|E_s$ and $DT^{-1}|E_u$ are contractions. Smale [@Smale67 Theorem 6.2] proved that the non-wandering set of an Axiom A diffeomorphism $T$ is the disjoint union of finitely many *basic sets* which 