Historic Behaviour for Random Expanding Maps on the Circle

TL;DR

This paper demonstrates that historic behaviour in expanding circle maps persists under small random perturbations, using a random Markov partition and a random version of Shub's Theorem.

Contribution

It introduces a random Markov partition and extends Shub's Theorem to show the stability of historic behaviour under stochastic perturbations.

Findings

Historic behaviour is preserved under small random perturbations.

A random Markov partition is constructed for perturbed maps.

A random version of Shub's Theorem is established.

Abstract

Takens constructed a residual subset of the state space consisting of initial points with historic behaviour for expanding maps on the circle. We prove that this statistical property of expanding maps on the circle is preserved under small random perturbations. The proof is given by establishing a random Markov partition, which follows from a random version of Shub's Theorem on topological conjugacy with the folding maps.