Dynamical Systems on Compact Metrizable Groups

Abstract

This paper is aim to extend Kenneth R. Berg's findings on the maximal entropy theorem and the ergodicity of measure convolution to the case of surjective homomorphisms. We further explores dynamical systems under surjective homomorphism in detail, especially the variation of entropy. Let $G_{1} $ and $G_{2} $ be compact metrizable groups, and suppose that $G_{2} $ acts freely on $G_{1} $, the continuous mapping $T_{1} $ and homomorphism $T_{2} :G_{2} \to G_{2} $ satisfy $T_{1} (yx)=T_{2} (y)T_{1} (x)$, where $y\in G_{2} ,{\rm \; }x\in G_{1} $. If $\mu _{0} \in M(T_{0} )$, $\mu _{0} '$ is the Haar extention of $\mu _{0} $, we proved that when $\mu \in $$M(T_{1} ,\mu _{0} )$, the entropy $h(T_{1} ,\mu _{0} '){\rm \; }$is always greater than or equal to $h(T_{1} ,\mu )$; if $\mu _{0} '$ is ergodic with respect to $T_{1} $, and the Haar measure $m$ on $G_{2} $ is ergodic with respect to $T_{2} $, and if $h(T_{1} ,\mu _{0} ')<\infty $, then the entropy $h(T_{1} ,\mu _{0} '){\rm \; }$is greater than $h(T_{1} ,\mu ).$ Finally, this paper also specifically discusses the ergodicity of the convolution of invariant measures. Let $T$ be a surjective homomorphism on $G$, if $(G,T,{\rm {\mathcal F}},\mu )$ and$(G,T,{\rm {\mathcal F}},\nu )$ are disjoint ergodic dynamical systems, then $\mu *\nu $ is ergodic. Via a proof by contradiction, the study demonstrates that the measure convolution of two disjoint ergodic dynamical systems can maintain ergodicity under the condition that $T$is a surjective homomorphism on $G$.