Semi-uniform ergodic theorems and applications to forced systems

Abstract

In nonlinear dynamics an important distinction exists between uniform bounds on growth rates, as in the definition of hyperbolic sets, and non-uniform bounds as in the theory of Liapunov exponents. In rare cases, for instance in uniquely ergodic systems, it is possible to derive uniform estimates from non-uniform hypotheses. This allowed one of us to show in a previous paper that a strange non-chaotic attractor for a quasiperiodically forced system could not be the graph of a continuous function. This had been a conjecture for some time. In this paper we generalize the uniform convergence of time averages for uniquely ergodic systems to a broader range of systems. In particular, we show how conditions on growth rates with respect to all the invariant measures of a system can be used to derive one-sided uniform convergence in both the Birkhoff and the sub-additive ergodic theorems. We apply the latter to show that any strange compact invariant set for a quasiperiodically forced system must support an invariant measure with a non-negative maximal normal Liapunov exponent; in other words, it must contain some `non-attracting' orbits. This was already known for the few examples of strange non-chaotic attractors that have rigorously been proved to exist. Finally, we generalize our semi-uniform ergodic theorems to arbitrary skew product systems and discuss the application of such extensions to the existence of attracting invariant graphs.