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Continuity of asymptotic entropy on free solvable groups

Eduardo Silva

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 $m\ge 2$.

Continuity of asymptotic entropy on free solvable groups

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 of rank and derived length .
Paper Structure (14 sections, 20 theorems, 37 equations)

This paper contains 14 sections, 20 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgements
  4. Preliminaries
  5. Random walks on groups and entropy
  6. Escape probability and the asymptotic range
  7. Convergence of probability measures
  8. Iterated wreath products and free solvable groups
  9. The coarse trajectory
  10. Free solvable groups
  11. Base groups with linear or quadratic growth
  12. Continuity of the escape probability on $\mathbb{Z}$ and $\mathbb{Z}^2$
  13. Continuity of asymptotic entropy on wreath products over $\mathbb{Z}$ or $\mathbb{Z}^2$
  14. Examples of discontinuity of asymptotic entropy

Key Result

Theorem 1.3

Let $S_{d,m}$ be the free solvable group of rank $d\ge 3$ and derived length $m\ge 2$. Let $\mu$ be a non-degenerate probability measure on $S_{d,m}$ with $H(\mu)<\infty$. Let $\{\mu_k\}_{k\ge 1}$ be a sequence of non-degenerate probability measures on $S_{d,m}$ such that $\lim_{k\to \infty}\mu_k(g)

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: Silva2025
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 29 more