Compact global attractors of discrete inclusions

The paper is dedicated to the study of the problem of the existence of compact global attractors of discrete inclusions and to the description of its structure. We consider a family of continuous mappings of a metric space WW into itself, and (W,fi)i∈I(W,fi)i∈I is the family of discrete dynamical systems. On the metric space WW we consider a discrete inclusion equation(1)ut+1∈F(ut)ut+1∈F(ut) associated with M≔{fi:i∈I}M≔{fi:i∈I}, where F(u)={f(u):f∈M}F(u)={f(u):f∈M} for all u∈Wu∈W. We give sufficient conditions (the family of maps MM is contracting in the extended sense) for the existence of a compact global attractor of (1). If the family MM consists of a finite number of maps, then the corresponding compact global attractor is chaotic. We study this problem in the framework of non-autonomous dynamical systems (cocyles).