Expansive Automorphisms on Locally Compact Groups

Abstract: We show that any connected locally compact group which admits an expansive automorphism is nilpotent. We also show that for any locally compact group $G$, $\alpha\in {\rm Aut}(G)$ is expansive if and only if for any $\alpha$-invariant closed subgroup $H$ which is either compact or normal, the restriction of $\alpha$ to $H$ is expansive and the quotient map on $G/H$ corresponding to $\alpha$ is expansive. We get a structure theorem for locally compact groups admitting expansive automorphisms. We prove that an automorphism on a non-discrete locally compact group can not be both distal and expansive.