  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 $G$.
