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Cyclicity, hypercyclicity and randomness in self-similar groups

Jorge Fariña-Asategui

TL;DR

The paper develops a framework connecting self-similar and automata group structures with measure-preserving dynamics on profinite groups endowed with Haar measure. It establishes that fractal self-similar groups yield measure-preserving dynamics, while super strongly fractal groups are ergodic and strongly mixing, enabling the use of ergodic theory to study randomness in these groups. A key result is that Haar-random elements in super strongly fractal profinite groups are hypercyclic with probability one, proven via a Birkhoff-type theorem for free semigroup actions. The work provides concrete findings for classical automata groups (Grigorchuk, Basilica, BSV) and gives contraction-based criteria ensuring cyclicity of non-finitary automorphisms, illustrating deep connections between fractality, dynamical properties, and the self-similar spectrum of these groups.

Abstract

We introduce the concept of cyclicity and hypercyclicity in self-similar groups as an analogue of cyclic and hypercyclic vectors for an operator on a Banach space. We derive a sufficient condition for cyclicity of non-finitary automorphisms in contracting discrete automata groups. In the profinite setting we prove that fractal profinite groups may be regarded as measure-preserving dynamical systems and derive a sufficient condition for the ergodicity and the mixing properties of these dynamical systems. Furthermore, we show that a Haar-random element in a super strongly fractal profinite group is hypercyclic almost surely as an application of Birkhoff's ergodic theorem for free semigroup actions.

Cyclicity, hypercyclicity and randomness in self-similar groups

TL;DR

The paper develops a framework connecting self-similar and automata group structures with measure-preserving dynamics on profinite groups endowed with Haar measure. It establishes that fractal self-similar groups yield measure-preserving dynamics, while super strongly fractal groups are ergodic and strongly mixing, enabling the use of ergodic theory to study randomness in these groups. A key result is that Haar-random elements in super strongly fractal profinite groups are hypercyclic with probability one, proven via a Birkhoff-type theorem for free semigroup actions. The work provides concrete findings for classical automata groups (Grigorchuk, Basilica, BSV) and gives contraction-based criteria ensuring cyclicity of non-finitary automorphisms, illustrating deep connections between fractality, dynamical properties, and the self-similar spectrum of these groups.

Abstract

We introduce the concept of cyclicity and hypercyclicity in self-similar groups as an analogue of cyclic and hypercyclic vectors for an operator on a Banach space. We derive a sufficient condition for cyclicity of non-finitary automorphisms in contracting discrete automata groups. In the profinite setting we prove that fractal profinite groups may be regarded as measure-preserving dynamical systems and derive a sufficient condition for the ergodicity and the mixing properties of these dynamical systems. Furthermore, we show that a Haar-random element in a super strongly fractal profinite group is hypercyclic almost surely as an application of Birkhoff's ergodic theorem for free semigroup actions.

Paper Structure

This paper contains 16 sections, 17 theorems, 32 equations, 1 figure.

Table of Contents

  1. introduction
  2. Preliminaries
  3. The Haar measure on profinite groups
  4. Measure-preserving dynamical systems
  5. Self-similar groups
  6. Cyclic and hypercyclic automorphisms
  7. The group $A$ and the space $V$
  8. Cyclicity and hypercyclicity in self-similar profinite groups
  9. Cyclicity in self-similar discrete groups
  10. First Grigorchuk group
  11. Basilica group
  12. Brunner-Sidki-Vieira group
  13. Automata groups
  14. Dynamical systems and fractality
  15. Fractality and measure-preserving dynamical systems
  16. ...and 1 more sections

Key Result

Theorem A

Let $G\le \mathrm{Aut}~T$ be a self-similar profinite group:

Figures (1)

  • Figure 1: From left to right, the Grigorchuk and the Basilica automata with the fully connected non-finitary states in red.

Theorems & Definitions (29)

  • Theorem A
  • Theorem B
  • Corollary 2
  • Corollary 2.2
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Definition 3.3: Cyclicity and hypercyclicity in self-similar profinite groups
  • Lemma 3.4
  • ...and 19 more