Characterization of the set of zero-noise limits measures of perturbed cellular automata

Abstract

We add small random perturbations to a cellular automaton and consider the one-parameter family $(F_\epsilon)_{\epsilon>0}$ parameterized by $\epsilon$ where $\epsilon>0$ is the level of noise. The objective of the article is to study the set of limiting invariant distributions as $\epsilon$ tends to zero denoted $\mathcal{M}_0^l$. Some topological obstructions appear, $\mathcal{M}_0^l$ is compact and connected, as well as combinatorial obstructions as the set of cellular automata is countable: $\mathcal{M}_0^l$ is $\Pi_3$-computable in general and $\Pi_2$-computable if it is uniformly approached. Reciprocally, for any set of probability measures $\mathcal{K}$ which is compact, connected and $\Pi_2$-computable, we construct a cellular automaton whose perturbations by an uniform noise admit $\mathcal{K}$ as the zero-noise limits measure and this set is uniformly approached. To finish, we study how the set of limiting invariant measures can depend on a bias in the noise. We construct a cellular automaton which realizes any connected compact set (without computable constraints) if the bias is changed for an arbitrary small value. In some sense this cellular automaton is very unstable with respect to the noise.