Dynamical Systems on Compact Metrizable Groups

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 G₁ and G₂ be compact metrizable groups, and suppose that G₂ acts freely on G₁, the continuous mapping T₁ and homomorphism T₂ :G₂ → G₂ satisfy T₁ (yx)=T₂ (y)T₁ (x), where y∈ G₂ , \; x∈ G₁. If μ ₀ ∈ M(T₀ ), μ ₀ ' is the Haar extention of μ ₀, we proved that when μ ∈M(T₁ ,μ ₀ )$, the entropy $h(T₁ ,μ ₀ '){ \; }$is always greater than or equal to $h(T₁ ,μ )$; if $μ ₀ '$ is ergodic with respect to $T₁ $, and the Haar measure $m$ on $G₂ $ is ergodic with respect to $T₂ $, and if $h(T₁ ,μ ₀ ')<∞ $, then the entropy $h(T₁ ,μ ₀ '){ \; }$is greater than $h(T₁ ,μ ).$ Finally, this paper also specifically discusses the ergodicity of the convolution of invariant measures. Let $T$ be a surjective homomorphism on $G$, if $(G,T,{ F},μ )$ and$(G,T,{ F},ν )$ are disjoint ergodic dynamical systems, then $μ *ν $ 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 $G$.