Sharkovskiis theorem under small random perturbations

TL;DR

This paper extends Sharkovskii's theorem to certain random dynamical systems using topological methods, enabling precise detection of random periodic points and periods under small perturbations.

Contribution

It introduces a topological approach via the random Conley index to establish a Sharkovskii-type theorem for perturbed one-dimensional maps, improving upon measure-theoretic methods.

Findings

Established a Sharkovskii-type theorem for random systems

Detected random periodic points with exact minimal periods

Constructed random periodic orbits for perturbed tent and logistic maps

Abstract

We establish a Sharkovskii-type theorem for a class of discrete random dynamical systems via the random Conley index. Using the continuation property of the Conley index, we extend classical forcing results to random systems obtained from small random perturbations of one-dimensional maps. In contrast to earlier measure-theoretic results, which are typically subject to an inherent period-doubling ambiguity (realizing period $n$ or $2n$), our topological approach allows us to detect random periodic points and orbits with precise minimal periods. This yields realisation results for arbitrary finite tails of the Sharkovskii ordering. These results are illustrated by constructing random periodic orbits for perturbed versions of the tent map and the logistic map.