Dense area-preserving homeomorphisms have zero Lyapunov exponents

Details Dense area-preserving homeomorphisms have zero Lyapunov exponents Dense area-preserving homeomorphisms have zero Lyapunov exponents

2010-04-05

arxiv.org/abs/1004.0641v2

Discrete and Continuous Dynamical Systems - A, vol 32, 4, 1231-1244, 2012

Mathematics

Dynamical Systems

13 pages. This version corrects an error in previous version. Specifically, there was a gap in the proof that the function "in

Scientific paper

We give a new definition for a Lyapunov exponent (called new Lyapunov exponent) associated to a continuous map. Our first result states that these new exponents coincide with the usual Lyapunov exponents if the map is differentiable. Then, we apply this concept to prove that there exists a C0-dense subset of the set of the area-preserving homeomorphisms defined in a compact, connected and boundaryless surface such that any element inside this residual subset has zero new Lyapunov exponents for Lebesgue almost every point. Finally, we prove that the function that associates an area-preserving homeomorphism, equipped with the C0-topology, to the integral (with respect to area) of its top new Lyapunov exponent over the whole surface cannot be upper-semicontinuous.

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