Ergodic Theory and Dynamical Systems



On the convergence to equilibrium states for certain non-hyperbolic systems


MICHIKO YURI a1
a1 Department of Business Administration, Sapporo University, Nishioka, Toyohira-ku, Sapporo 062, Japan

Abstract

We study the convergence to equilibrium states for certain non-hyperbolic piecewise invertible systems. The multi-dimensional maps we shall consider do not satisfy Renyi's condition (uniformly bounded distortion for any iterates) and do not necessarily satisfy the Markov property. The failure of both conditions may cause singularities of densities of the invariant measures, even if they are finite, and causes a crucial difficulty in applying the standard technique of the Perron–Frobenius operator. Typical examples of maps we consider admit indifferent periodic orbits and arise in many contexts. For the convergence of iterates of the Perron–Frobenius operator, we study continuity of the invariant density.

(Received March 29 1995)