Normal Lyapunov exponents and asymptotically stable attractors

Abstract

Suppose f is a C ^1+α map and leaves a lower-dimensional compact attractor A. In this article, we show that if for every f-ergodic probability measure supported on A, the normal Lyapunov exponents are negative, then this attractor could be a high-dimensional attractor. Moreover, we prove that the supremum of the normal Lyapunov exponents on the set of all ergodic measures can be achieved.

Keywords:

• Normal Lyapunov Exponents
• Asymptotically Stable Attractors
• Ergodic Measure

2000 Mathematics Subject Classifications:

• Primary 34D35
• Secondary 58F12

Acknowledgements

The author would like to thank anonymous referees for their helpful comments and suggestions which led to an important improvement of original manuscript. Cao is partially supported by NSFC(10571130), NCET and 973 Project (2007CB814800).