# American Institute of Mathematical Sciences

May  2018, 23(3): 1199-1217. doi: 10.3934/dcdsb.2018148

## Random Delta-Hausdorff-attractors

 1 Technische Universität Berlin, Fak. Ⅱ, Institut für Mathematik, Sekr. MA 7-5, Straße des 17. Juni 136,10623 Berlin, Germany

Received  March 2017 Revised  August 2017 Published  May 2018 Early access  February 2018

Fund Project: The second author is supported by the DFG-SPP 1590.

Global random attractors and random point attractors for random dynamical systems have been studied for several decades. Here we introduce two intermediate concepts: Δ-Hausdorff-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-Hausdorff-attractors for different values of Δ. It seems that both concepts are new even in the context of deterministic dynamical systems.

Citation: Michael Scheutzow, Maite Wilke-Berenguer. Random Delta-Hausdorff-attractors. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1199-1217. doi: 10.3934/dcdsb.2018148
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