Piecewise contraction maps and its applications

Abstract:

Our talk concerns dynamical systems which are defined by piecewise contraction maps (PC maps). There is a large literature which deals with the dynamical behavior of PC maps defined on convex subsets of Euclidean spaces in different contexts. Our aim is to show that, under certain conditions, a typical PC map, in the measure theoretical sense of the parameter space, is asymptotically periodic which means that the map has finitely many periodic orbits and every orbit converges to a periodic orbit. Our setup is the following: We fix an Iterated Function System {φ1, . . . , φn}, where each map φi : X → X is a contraction. Next we use a suitable parameter space and associate to each parameter a partition X1, . . . , Xn of X. In this way, each parameter defines a piecewise contraction map f : X → X given by f(x) = φi(x) for every x ∈ Xi.

Posted on Apr 25, 2016 in Dynamical Systems, Seminars