COMPACT GLOBAL ATTRACTORS OF DISCRETE CONTROL SYSTEMS DAVID CHEBAN AND CRISTIANA MAMMANA Abstract. The paper is dedicated to the study of the problem of existence of compact global attractors of discrete control systems and to description of its structure. We consider a family of continuous mappings of metric space W into itself, and (W, f i ) i∈I is the family of discrete dynamical systems. On the metric space W we consider a discrete inclusion (1) u t+1 ∈ F (ut ) associated by M := {f i : i ∈ I }, where F (u)= {f (u) : f ∈ M} for all u ∈ W. We give the sufficient conditions (the family of maps M is contracting in the extended sense) of existence of compact global attractor of (1). If the family M consists a finite number of maps, then corresponding compact global attractor is chaotic. We study this problem in the framework of non- autonomous dynamical systems (cocyles). 1. Introduction The aim of this paper is the study of the problem of existence of compact global attractors of discrete control systems (see, for example, Bobylev, Emel’yanov and Korovin [6], Bobylev, Zalozhnev and Klykov [7], Emel’yanov, Korovin and Bobylev [21] and the references therein). Let W be a metric space, M := {f i : i ∈ I } be a family of continuous mappings of W into itself and (W,f i ) i∈I be the family of discrete dynamical systems, where (W,f ) is a discrete dynamical system generated by positive powers of continuous map f : W → W . On the space W we consider a discrete inclusion u t+1 ∈ F (u t ) associated by M := {f i : i ∈ I } (DI (M)), where F (u)= {f (u): f ∈ M} for all u ∈ W. A solution of discrete inclusion DI (M) is called (see, for example, [6, 21, 25]) a sequence {{x j }| j ≥ 0}⊂ W such that x j = f ij x j−1 for some f ij ∈M (trajectory of DI (M)), i.e. x j = f ij f ij-1 ...f i1 x 0 all f i k ∈M. Date : April 14, 2004. 1991 Mathematics Subject Classification. 37B25, 37B55, 39A11, 39C10, 39C55. Key words and phrases. Global attractor; set-valued dynamical system; control system, chaotic attractor, collage, cocycle. 1