Cyclicity, hypercyclicity and randomness in self-similar groups

TL;DR

This paper explores the concepts of cyclicity and hypercyclicity in self-similar groups, establishing conditions for these properties and demonstrating their implications for dynamical systems and randomness.

Contribution

It introduces cyclicity and hypercyclicity in self-similar groups, providing new criteria and linking these properties to ergodic theory and random elements.

Findings

Non-finitary automorphisms can be cyclic under certain conditions.

Fractal profinite groups exhibit measure-preserving dynamical behavior.

Haar-random elements in super strongly fractal groups are hypercyclic almost surely.

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.