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

TL;DR

This paper investigates how perturbations affect the uniqueness of invariant measures in continuous maps on the Cantor set, showing that the set of perturbation probabilities with unique measures is a G_delta set containing 1, and generically equals (0,1].

Contribution

It characterizes the sets of perturbation probabilities for which perturbed maps have unique measures, revealing their topological structure and generic properties.

Findings

The set of epsilon with unique measure is a G_delta set containing 1.

Generically, this set is the interval (0,1].

The structure of these sets is fully characterized.

Abstract

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 $]0, 1]$.