Flexibility versus genericity of phase diagrams of perturbed continuous maps on the Cantor set

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 $]0,1]$.