The question is in the title really.
Does topological conjugacy preserve expansiveness? That is if two topological dynamical systems $f$ and $ g$ are topologically conjugate, and $f$ is expansive, must $g$ be also expansive?
(Expansive: a topological dynamical system $f$ on a metric space $(X, d)$ is called expansive if there exists $c > 0$ such that for each $x \neq y \in X$, there exists $n \in \mathbb{N}$, so that $ $ $d\left(f^{n}(x), f^{n}(y)\right) \geq c$.)
I cannot find any counterexample, so I would think yes. Is this the case? How to prove this if so?
Assume that $X$ is compact. Then this should follow relatively easily from uniform continuity of the conjugation map $h$.
If $X$ is not compact, here is a counter example: consider $f(x)=g(x)=x/2$ on $X=(0,1]$. You can consider two distances inducing the same topology on $X$, $d_1(x,y)=|x-y|$ and $d_2(x,y)=|\frac{1}{x}-\frac{1}{y}|$. The maps $f$ and $g$ (acting respectively on $(X,d_1)$ and $(X,d_2)$) are conjugated by the identity, which is a homeomorphism since $d_1$ and $d_2$ define the same topology. But $g$ is expansive on $(X,d_2)$ while $f$ is not on $(X,d_1)$.