Symplectomorphisms with positive metric entropy
Copyright policy )Symplectomorphisms with positive metric entropy
We obtain a dichotomy for $C^1$-generic symplectomorphisms: either all the Lyapunov exponents of almost every point vanish, or the map is partially hyperbolic and ergodic with respect to volume. This completes a program first put forth by Ricardo Mañé. A main ingredient in our proof is a generalization to partially hyperbolic invariant sets of the main result in [Dolgopyat-Wilkinson] that stable accessibility is $C^1$ dense among partially hyperbolic diffeomorphisms.
- French National Centre for Scientific Research France
- University of Paris-Saclay France
- Département de Mathématiques France
- University of Chicago United States
- University of Zurich Switzerland
General Mathematics, Lyapunov exponents, Dynamical Systems (math.DS), symplectomorphisms, 10123 Institute of Mathematics, 510 Mathematics, Symplectic and canonical mappings, Dynamical systems with hyperbolic orbits and sets, Dynamical systems involving smooth mappings and diffeomorphisms, FOS: Mathematics, Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.), Mathematics - Dynamical Systems, Partially hyperbolic systems and dominated splittings, 2600 General Mathematics, partially hyperbolic diffeomorphisms
General Mathematics, Lyapunov exponents, Dynamical Systems (math.DS), symplectomorphisms, 10123 Institute of Mathematics, 510 Mathematics, Symplectic and canonical mappings, Dynamical systems with hyperbolic orbits and sets, Dynamical systems involving smooth mappings and diffeomorphisms, FOS: Mathematics, Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.), Mathematics - Dynamical Systems, Partially hyperbolic systems and dominated splittings, 2600 General Mathematics, partially hyperbolic diffeomorphisms
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