Statistical Stability for Multi-Substitution Tiling Spaces

TL;DR

This paper studies the stability of frequencies in multi-substitution tiling spaces, showing that ergodic limits converge at a certain rate and depend continuously on the substitution sequence.

Contribution

It introduces a framework for analyzing the convergence rate and continuity of ergodic limits in multi-substitution tiling spaces.

Findings

Ergodic limits of patch frequencies converge at a quantifiable rate.

These limits vary continuously with the substitution sequence.

The associated dynamical systems are uniquely ergodic.

Abstract

Given a finite set ${S_1...,S_k}$ of substitution maps acting on a certain finite number (up to translations) of tiles in $\rd$, we consider the multi-substitution tiling space associated to each sequence $\bar a\in {1,...,k}^{\mathbb{N}}$. The action by translations on such spaces gives rise to uniquely ergodic dynamical systems. In this paper we investigate the rate of convergence for ergodic limits of patches frequencies and prove that these limits vary continuously with $\bar a$.