On the Besicovitch-Stability of Noisy Random Tilings

TL;DR

This paper investigates the stability of symbolic dynamical systems called SFTs under noise, providing a full classification in one dimension and stability results for higher dimensions and specific aperiodic tilings.

Contribution

It introduces a noisy framework for SFTs, classifies stability in 1D, and proves stability under Bernoulli noise in higher dimensions, extending to aperiodic tilings.

Findings

Complete classification of stability in 1D SFTs

Stability under Bernoulli noise in higher-dimensional periodic SFTs

Extension of stability results to an aperiodic tiling example

Abstract

In this paper, we introduce a noisy framework for SFTs, allowing some amount of forbidden patterns to appear. Using the Besicovitch distance, which permits a global comparison of configurations, we then study the closeness of noisy measures to non-noisy ones as the amount of noise goes to 0. Our first main result is the full classification of the (in)stability in the one-dimensional case. Our second main result is a stability property under Bernoulli noise for higher-dimensional periodic SFTs, which we finally extend to an aperiodic example through a variant of the Robinson tiling.