On Periodic Points for Systems of Weakly Coupled 1-Dim Maps

In paper we present the topological method of proving the existence of periodic in multidimensional nonuniformly hyperbolic dynamical systems. We apply our method to the maps of the form F (x1 ; : : : ; xn) := (f(x1); f(x2); : : : ; f(xn)) +G : R n ! R n where f : R! R is either logistic or tent map in the chaotic region and G is sufficiently small in C 0 norm. We prove that F have inifinite number periodic points with symbolic dynamics. We prove also the result for spatial distribution of periodic points for F in the invariant set. Another considered example is Rossler walking stick diffeomorphism, for which we prove the existence of inifinite number of periodic points. Keywords: fixed point index, chaos, nonuniformly hyperbolic systems AMS classification scheme numbers: 58F15, 58G10 1 Introduction The aim of this paper is to develop the method of TS-maps (topological shifts) introduced by the author in [?] for proving existence of infinite number of periodic points in multidi..