Consider the dynamical system constitued by a continuous function $F:\mathcal{ A}^\mathbb{ N}\to\mathcal{ A}^\mathbb{ N}$ where $\mathcal{ A}$ is a finite alphabet. The perturbed counterpart, denoted by $F_\epsilon$, is obtained after each iteration of $F$ by modifying each cell independently with probability $\epsilon\in[0,1]$ and choosing the new value uniformly. We characterize the possible sets of $\epsilon\in[0,1]$ such that $F_\epsilon$ has a unique measure. These sets are exactly the $G_\delta$ sets (countable intersection of open sets) of $[0, 1]$ which contain 1. However, we show that generically this set is $]0, 1]$.
Categories: math.DS