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Necessary and sufficient conditions for semi-uniform ergodic theorems and their application. (English) Zbl 1110.37016

Summary: It has been established one-side uniform convergence in both the Birkhoff and sub-additive ergodic theorems under conditions on growth rates with respect to all the invariant measures. In this paper, we show that these conditions are both necessary and sufficient. These results are applied to study quasiperiodically forced systems. Some meaningful geometric properties of invariant sets of such systems are presented. We also show that any strange compact invariant set of a \({\mathcal C}^1\) quasiperiodically forced system must support an invariant measure with a nonnegative normal Lyapunov exponent.

MSC:

37C55 Periodic and quasi-periodic flows and diffeomorphisms
28D05 Measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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