Covering relations and Lyapunov condition for topological conjugacy

Abstract

In this paper, we study stability of non-autonomous discrete dynamical systems. For a two-sided non-autonomous systems with covering relations determined by a transition matrix A, we show that any small C0 perturbed system has a sequence of compact invariant sets restricted to which the system is topologically semi-conjugate to σA, the two-sided subshift of finite type induced by A. Together with Lyapunov condition of good rate, the semi-conjugacy will become conjugacy. Moreover, if the Lyapunov condition is strict and has perfect rate, then any small C1 perturbed systems is topological conjugate to σA. We also study topological chaos of one-sided systems and systems with limit functions. Lack of hyperbolicity and the time dependence of the rate prevent us from applying classical hyperbolic results or earlier works for autonomous systems: cone condition and Lyapunov function. Two examples are provided to demonstrate the existence of a non-trivial, non-hyperbolic invariant and essential of controlling rate.