Random Delta-Hausdorff-attractors

Global random attractors and random point attractors for random dynamical systems have been studied for several decades. Here we introduce two intermediate concepts: $\mathrm{\Delta }$-attractors are characterized by attracting all deterministic compact sets of Hausdorff dimension at most $\mathrm{\Delta }$, where $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$-attractors for different values of $\mathrm{\Delta }$. It seems that both concepts are new even in the context of deterministic dynamical systems.