Let $G$ be a Hausdorff locally compact group and let $\alpha$ be an automorphism of $G$. Say $\alpha$ is *(forwards) topologically recurrent* if for all $g \in G$ and all neighbourhoods $O$ of $g$, the sequence $(\alpha^n(g))_{n \ge 0}$ returns to $O$ infinitely often.

For instance, if $G$ is a profinite group then every inner automorphism is topologically recurrent. On the other hand if $\alpha^n(g) \rightarrow 1$ for some $g \not= 1$, then $\alpha$ is not topologically recurrent.

Is there a standard name for this property or for something similar? Has anyone worked on these?

A stronger property would be if $\alpha^n(g)$ visits every open set infinitely often, but I don't want to assume this. (For instance, I want to allow distal automorphisms, that is automorphisms where every non-trivial orbit is isolated from the identity.)