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

Ming-Chia Li*, Ming Jiea Lyu

*Corresponding author for this work

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

In this paper, we study topological dynamics of high-dimensional systems which are perturbed from a continuous map on Rm×Rk of the form (f(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 C0 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 C1 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 Zgliczyński (2009) [11]. Our results extend those in Juang et al. (2008) [1], Li et al. (2008) [2], Li and Malkin (2006) [3], Misiurewicz and Zgliczyński (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 Hénon families.

Original languageEnglish
Pages (from-to)799-812
Number of pages14
JournalJournal of Differential Equations
Volume250
Issue number2
DOIs
StatePublished - 15 Jan 2011

Keywords

  • Covering relation
  • Liapunov condition
  • Multidimensional perturbation
  • Topological dynamics

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