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.

翻译：本文对元胞自动机施加微小随机扰动，并考虑由噪声水平参数 $\epsilon>0$ 参数化的单参数族 $(F_\epsilon)_{\epsilon>0}$。文章的目标是研究当 $\epsilon$ 趋于零时极限不变分布的集合，记为 $\mathcal{M}_0^l$。该集合存在一些拓扑障碍：$\mathcal{M}_0^l$ 是紧致且连通的；同时由于元胞自动机构成的集合是可数的，也存在组合障碍：一般情况下 $\mathcal{M}_0^l$ 是 $\Pi_3$-可计算的，若其能被一致逼近则为 $\Pi_2$-可计算。反之，对于任意紧致、连通且 $\Pi_2$-可计算的概率测度集 $\mathcal{K}$，我们构造一个元胞自动机，使其在均匀噪声扰动下以 $\mathcal{K}$ 作为零噪声极限测度集，且该集合能被一致逼近。最后，我们研究极限不变测度集如何依赖于噪声中的偏差。我们构造了一个元胞自动机，当噪声偏差发生任意微小变化时，它能实现任意连通紧致集（无需可计算性约束）。从某种意义上说，该元胞自动机对噪声具有高度不稳定性。