Ergodicity of power series-map on the simplex of group algebra of a finite group

Abstract

A finite group $G$, its group algebra $R[G]$ over the field of real numbers, any power series $p(t)=a_0+a_{1}t+a_2{t}^2+...$, where $a_i\ge 0$, and $a_0+a_1+a_2+...=1$, and simplex $$S=\{x=\sum _{g\in G}{x}_{g}g\in R[G]:\sum _{g\in G}{x}_g=1,x_g\ge 0foranyg\in G\}$$ are considered. Ergodicity of the map $p:S\to S$, where $p(x)=a_0+a_{1}x+a_2{x}^2+>...$ for $x\in S$, on $S$ is shown. The regularity of this map at a given point $x\in S$ is investigated as well.