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, an automorphism α of G is expansive if and only if for any α-invariant closed subgroup H which is either compact or normal, the restriction of α to H is expansive and the quotient map on G/H corresponding to α is expansive. We get a structure theorem for locally compact groups admitting expansive automorphisms. We prove that an automorphism of a non-discrete locally compact group can not be both distal and expansive.

The author would like to thank Helge Glöckner for extensive discussions and the Fields Institute, Toronto, Canada for hospitality while some part of the work was done as a visiting scientific researcher to the Theme Period on Group Structure, Group Actions and Ergodic Theory in February 2014. The author is grateful to the referee for comments and suggestions which led to a significant improvement in the presentation of the manuscript.