# American Institue of Mathematical Sciences

2006, 14(3): 409-417. doi: 10.3934/dcds.2006.14.409

## Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications

 1 Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, China 2 School of Mathematical Sciences, Peking University, Beijing, 100871, China 3 School of Sciences, Beihang University, Beijing, 100083, China

Received  July 2004 Revised  September 2005 Published  December 2005

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 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 non-negative normal Lyapunov exponent.
Citation: Zuohuan Zheng, Jing Xia, Zhiming Zheng. Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 409-417. doi: 10.3934/dcds.2006.14.409
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