The weak Smale horseshoe and mean hyperbolicity

Abstract

The paper concerns the mean hyperbolicity of the skew-product flows induced by the perturbation equations driven by varieties of non-periodic forcings. The weakly averaged horseshoe can be constructed as a mean hyperbolic invariant set in the Poincare section for high dimensional phase space. Due to the non-periodic property, the Poincare return map restricted to the weakly averaged horseshoe region can semi-conjugate to the full Bernoulli shift on infinite symbols, which implies the infinitely many independent choices on the length of return time. As a direct application of mean hyperbolicity, we extend the shadowing lemma due to Liao to the general nonautonomous dynamical systems.