Expansive homeomorphism

Remarks

On a compact metric space, expansiveness does not depend on the choice of a metric that gives the topology, although the constant does. Example: the two-sided full shift $\sigma$ on $\set{0, 1}^{\Z}$ with $d(x, y) = 2^{-\min\set{\abs{i} : x_i \neq y_i}}$ and $c = 1/2$. If $x \neq y$, pick $i$ with $x_i \neq y_i$; then $\sigma^i(x)$ and $\sigma^i(y)$ differ at coordinate $0$, so their distance is $1 > 1/2$. Non-example: an isometry such as the rotation $x \mapsto x + \alpha \bmod 1$ of the circle. For any $c > 0$, two distinct points at distance less than $c$ stay at that distance for all $n \in \Z$. An expansive homeomorphism of a compact metric space has finite topological entropy.