Substitution-dynamics and invariant measures for infinite alphabet-path space

TL;DR

This paper investigates substitution systems on infinite alphabets as Borel dynamical systems, constructing models and invariant measures to understand their complex dynamics.

Contribution

It introduces generalized Bratteli-Vershik models for infinite alphabet substitutions and provides a canonical method to construct shift-invariant measures.

Findings

Constructed stationary and non-stationary models for infinite alphabet substitutions.

Provided a new method to construct shift-invariant measures, including infinite measures.

Enhanced understanding of the dynamics of infinite alphabet substitution systems.

Abstract

We study substitutions on countably infinite alphabet (without compactification) as Borel dynamical systems. We construct stationary and non-stationary generalized Bratteli-Vershik models for a class of such substitutions, known as left determined. In this setting of Borel dynamics, using a stationary generalized Bratteli-Vershik model, we provide a new and canonical construction of shift-invariant measures (both finite and infinite) for the associated class of subshifts.