Continuity of asymptotic entropy on free solvable groups

TL;DR

This paper proves that the asymptotic entropy varies continuously with the step distribution for certain free solvable groups, under specific conditions on the measures.

Contribution

It establishes the continuity of asymptotic entropy as a function of the step distribution on free solvable groups with finite Shannon entropy.

Findings

Asymptotic entropy is continuous with respect to the step distribution.

Continuity holds for non-degenerate measures with finite Shannon entropy.

Results apply to free solvable groups of rank at least 3 and derived length at least 2.

Abstract

We prove the continuity of asymptotic entropy, as a function of the step distribution, among non-degenerate probability measures with finite Shannon entropy on the free solvable group $S_{d,m}$ of rank $d\ge 3$ and derived length $m\ge 2$.