Periodic measures and partially hyperbolic homoclinic classes
Copyright policy )Periodic measures and partially hyperbolic homoclinic classes
In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbation of the time one map of a transitive Anosov flow, or of the skew product of an Anosov torus diffeomorphism by rotation of the circle.
48 pages, 5 figures
- University of Paris-Sud France
- Peking University China (People's Republic of)
- University of Burgundy France
- Peking University China (People's Republic of)
- Institut de Mathématiques de Bourgogne France
Ergodicity, mixing, rates of mixing, [MATH] Mathematics [math], Dynamical Systems (math.DS), stability, Invariant manifold theory for dynamical systems, partial hyperbolicity, quasi-hyperbolic string, Blender, Dynamical systems with hyperbolic orbits and sets, non-hyperbolic measure, FOS: Mathematics, systems, Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics, Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.), Mathematics - Dynamical Systems, robust cycle, ergodic measure, Partially hyperbolic systems and dominated splittings, periodic measure, Smooth ergodic theory, invariant measures for smooth dynamical systems, Lyapunov exponent, foliations
Ergodicity, mixing, rates of mixing, [MATH] Mathematics [math], Dynamical Systems (math.DS), stability, Invariant manifold theory for dynamical systems, partial hyperbolicity, quasi-hyperbolic string, Blender, Dynamical systems with hyperbolic orbits and sets, non-hyperbolic measure, FOS: Mathematics, systems, Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics, Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.), Mathematics - Dynamical Systems, robust cycle, ergodic measure, Partially hyperbolic systems and dominated splittings, periodic measure, Smooth ergodic theory, invariant measures for smooth dynamical systems, Lyapunov exponent, foliations
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