Semi-uniform ergodic theorems and applications to forced systems

In nonlinear dynamics an important distinction exists between uniform bound s 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 fr om non-uniform hypotheses. This allowed one of us to show in a previous pap er that a strange non-chaotic attractor for a quasiperiodically forced syst em could not be the graph of a continuous function. This had been a conject ure for some time. In this paper we generalize the uniform convergence of t ime averages for uniquely egodic 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 conv ergence in both the Birkhoff and the sub-additive ergodic theorems. We appl y the latter to show that any strange compact invariant set for a quasiperi odically forced system must support an invariant measure with a non-negativ e maximal normal Liapunov exponent; in other words, it must contain some 'n on-attracting' orbits. This was already known for the few examples of stran ge non-chaotic attractors that have rigorously been proved to exist. Finall y, we generalize our semi-uniform ergodic theorems to arbitrary skew produc t systems and discuss the application of such extensions to the existence o f attracting invariant graphs.