DENSE AREA-PRESERVING HOMEOMORPHISMS HAVE ZERO LYAPUNOV EXPONENTS

作者：Bessa Mario^*; Silva Cesar M

来源：Discrete and Continuous Dynamical Systems, 2012, 32(4): 1231-1244.

DOI：10.3934/dcds.2012.32.1231

• Area-preserving homeomorphisms
• Lyapunov exponents
• topological dynamics

We give a new definition (different from the one in [14]) 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 C(0)-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 C(0)-topology, to the integral (with respect to area) of its top new Lyapunov exponent over the whole surface cannot be upper-semicontinuous.

• 出版日期2012-4