Topological dynamics for multidimensional perturbations of maps with covering relations and Liapunov condition

Abstract

In this paper, we study topological dynamics of high-dimensional systems which are perturbed from a continuous map on R(m) x R(k) of the form (1(x), g(x, y)). Assume that f has covering relations determined by a transition matrix A. If g is locally trapping, we show that any small C(0) perturbed system has a compact positively invariant set restricted to which the system is topologically semi-conjugate to the one-sided subshift of finite type induced by A. In addition, if the covering relations satisfy a strong Liapunov condition and g is a contraction, we show that any small C(1) perturbed homeomorphism has a compact invariant set restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by A. Some other results about multidimensional perturbations of f are also obtained. The strong Liapunov condition for covering relations is adapted with modification from the cone condition in Zgliczynski (2009) [11]. Our results extend those in Juang et al. (2008) [1], Li et al. (2008) [2]. Li and Malkin (2006) [3]. Misiurewicz and Zgliczynski (2001) [4] by considering a larger class of maps f and their multidimensional perturbations, and by concluding conjugacy rather than entropy. Our results are applicable to both the logistic and Henon families. (C) 2010 Elsevier Inc. All rights reserved.