Genericity of zero Lyapunov exponents

Abstract

We show that, for any compact surface, there is a residual (dense $G_{\delta }$) set of $C^1$ area preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponents a.e. This result was announced by R. Mane, but no proof was available. We also show that for any fixed ergodic dynamical system over a compact space, there is a residual set of continuous $SL(2,R)$-cocycles which either are uniformly hyperbolic or have zero exponents a.e.

28 pages, 1 figure. This is a revised, more readable, version of the preprint distributed in 2000

Cited by in corpus (80)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• A formula with some applications to the theory of Lyapunov exponents
• Lyapunov exponents for random perturbations of some area-preserving maps including the standard map
• Examples of Discontinuity of Lyapunov Exponent in Smooth Quasi-Periodic Cocycles
• Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts
• Some Characterizations of Domination
• Generic dynamics of 4-dimensional C2 Hamiltonian systems
• $L^p$-generic cocycles have one-point Lyapunov spectrum
• Variational equalities of entropy in nonuniformly hyperbolic systems
• Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2,R) cocycles
• Continuity of Lyapunov Exponents for Random 2D Matrices
• A survey on partially hyperbolic dynamics
• A remark on conservative diffeomorphisms
• $C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents
• Ergodic Properties of Invariant Measures for C^{1+α} nonuniformly hyperbolic systems
• Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems
• Continuity properties of Lyapunov exponents for surface diffeomorphisms
• Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle
• Discontinuity of Lyapunov Exponents Near Fiber Bunched Cocycles
• Hyperbolicity and recurrence in dynamical systems: a survey of recent results
• Dominated splitting and zero volume for incompressible three-flows
• Removing zero Lyapunov exponents in volume-preserving flows
• Tangent bundles dynamics and its consequences
• On the regularization of conservative maps
• Density of positive Lyapunov exponents for SL(2,R) cocycles
• Unstable entropy in smooth ergodic theory
• Generic singular continuous spectrum for ergodic Schrödinger operators
• Quasi-Periodic Schrödinger Cocycles with Positive Lyapunov Exponent are not Open in the Smooth Topology
• Continuity of Lyapunov exponents is equivalent to continuity of Oseledets subspaces
• Dynamics of $C^1$-diffeomorphisms: global description and prospects for classification
• Local perturbations of conservative $C^1$-diffeomorphisms
• $C^1$-openness of non-uniform hyperbolic diffeomorphisms with bounded $C^2$ norm
• On the abominable properties of the almost Mathieu operator with well approximated frequencies
• Parametric Furstenberg Theorem on Random Products of $SL(2,\mathbb{R})$ matrices
• Generic area-preserving reversible diffeomorphisms
• (Semi)continuity of the entropy of Sinai probability measures for partially hyperbolic diffeomorphisms
• Lower bounds on the Lyapunov exponents of stochastic differential equations
• A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations
• Sharp Holder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles
• A new proof of the Herman-Avila-Bochi formula for Lyapunov exponents of SL(2,R)-cocycles
• On $SL(2,\mathbb{R})$-cocycles over irrational rotations with secondary collisions
• Mixed Dynamics in a Parabolic Standard Map
• Continuity of Lyapunov exponents for non-uniformly fiber-bunched cocycles
• Positivity of the top Lyapunov exponent for cocycles on semisimple Lie groups over hyperbolic bases
• On topological genericity of the mode-locking phenomenon
• Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles
• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles
• Positive Lyapunov exponents for symplectic cocycles
• Center Lyapunov exponents in partially hyperbolic dynamics
• Non-Uniform hyperbolicity for infinite dimensional cocycles
• Abundance of elliptic dynamics on conservative 3-flows
• The Lyapunov exponents of generic volume preserving and symplectic systems
• Regularity of Lyapunov Exponents for Diffeomorphisms with Dominated Splitting
• Non-uniformly hyperbolic endomorphisms
• Density of mode-locking property for quasi-periodically forced Arnold circle maps
• Minimality and stable Bernouliness in dimension 3
• Stable ergodicity of dominated systems
• Hamiltonian elliptic dynamics on symplectic 4-manifolds
• Some advances on generic properties of the Oseledets splitting
• Perturbations of C1-diffeomorphisms and dynamics of generic conservative diffeomorphisms of surface
• The $C^1$ density of nonuniform hyperbolicity in $C^r$ conservative diffeomorphisms
• Approximation of Oseledets Splittings
• Non uniform hyperbolicity and elliptic dynamics
• Spatial discretizations of generic dynamical systems
• Hilbert Space Lyapunov Exponent stability
• A dichotomy in area-preserving reversible maps
• Imperfect friezes of integers
• On density of positive Lyapunov exponents for $C^1$ symplectic diffeomorphisms
• Singular analytic linear cocycles with negative infinite Lyapunov exponents
• Lyapunov Exponents of Linear Cocycles Over Markov Shifts
• Contributions to the Geometric and Ergodic Theory of Conservative Flows
• Center bunching without dynamical coherence
• Uniform gap in Lyapunov exponents and dominated splitting for linear cocycles
• On the entropy of conservative flows
• Some results on perturbations to Lyapunov exponents
• Continuity of Lyapunov exponents in all Hölder topologies for irreducible cocycles
• On the abundance of non-zero central Lyapunov exponents, physical measures and stable ergodicity for partially hyperbolic dynamics
• Fine properties of Lp-cocycles which allow abundance of simple and trivial spectrum
• Trivial and simple spectrum for SL(2,R) cocycles with free base and fiber dynamics
• Cocycles with Quasi-Conformality I: Stability and abundance