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

v1: 20 pages v2: 20 pages, corrected typos, streamlined proofs