Entropy for group endomorphisms and homogeneous spaces

Abstract

Topological entropy h d ( T ) {h_d}(T) is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare h d ( T ) {h_d}(T) with measure theoretic entropy h ( T ) h(T) ; in particular h ( T ) = h d ( T ) h(T) = {h_d}(T) for Haar measure and affine maps T T on compact metrizable groups. A particular case of this yields the well-known formula for h ( T ) h(T) when T T is a toral automorphism.