Random Delta-Hausdorff-attractors

TL;DR

This paper introduces intermediate concepts called $ ext{Delta}$-attractors and cc-attractors for random dynamical systems, which attract sets based on Hausdorff dimension and countability, respectively, expanding the understanding of attractors.

Contribution

It proposes new types of attractors, $ ext{Delta}$-attractors and cc-attractors, and demonstrates their existence and properties through examples, extending attractor theory.

Findings

$ ext{Delta}$-attractors depend on Hausdorff dimension $ ext{Delta}$

Existence of multiple $ ext{Delta}$-attractors for different $ ext{Delta}$ values

Both concepts are potentially new even in deterministic systems

Abstract

Global random attractors and random point attractors for random dynamical systems have been studied for several decades. Here we introduce two intermediate concepts: $\Delta$-attractors are characterized by attracting all deterministic compact sets of Hausdorff dimension at most $\Delta$, where $\Delta$ is a non-negative number, while cc-attractors attract all countable compact sets. We provide two examples showing that a given random dynamical system may have various different $\Delta$-attractors for different values of $\Delta$. It seems that both concepts are new even in the context of deterministic dynamical systems.