Genericity of zero Lyapunov exponents

By Jairo Bochi. Ergodic Theory and Dynamical Systems, 2002

267 citations recorded by the source corpus.

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. Mañé, 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,\mathbb{R})-cocycles which either are uniformly hyperbolic or have zero exponents a.e.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3f04090c538c18dfa9dab0e2fba6adb486cb965e

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Citations in the corpus

Nonuniform hyperbolicity for C1-generic diffeomorphisms (2008, methodology): “was proven by Bochi [ Boc1 ] in 2000, later extended to higher dimensions by Bochi and Viana [BocV], and flnally settled in the original (symplectic, in arbitrary dimension) statement of Ma~n¶e by Bochi [Boc2] in 2007.”
Every compact manifold carries a completely hyperbolic diffeomorphism (2002, methodology): “(2) If for any sufficiently small perturbation of f the subspaceE2 does not admit further splitting, then using results of Manẽ [M1] (see also [M2]) and Bochi [Bo] one can approximate f by a diffeomorphism g such that all Lyapunov exponents of g along E2 are close to each other.”
On the regularization of conservative maps (2008, methodology): “Zehnder’s Theorem has been used in [3] and [5] to achieve this conclusion in the symplectic case, and such a result is necessary for the conclusion of the central dychotomy of [3].”
Nonuniform hyperbolicity for C^1-generic diffeomorphisms (2008, background): “ 1The space Merg f ( ) of ergodic measures of a diﬀeomorphism f is a Baire space when endowed with the weak topology, so that its residual subsets are dense; see Subsection 5.1. 1 was proven by Bochi [Boc1] in 2000, later extended to higher dimensions by Bochi and Viana [BocV], and ﬁnally settled in”
Lyapunov exponents for random perturbations of some area-preserving maps including the standard map (2017, background): “he substantial body of work on 1D Schr¨odinger operators (e.g. [25, 12, 29, 2]). We also note the C1 genericity of zero Lyapunov exponents of volume-preserving surface diﬀeomorphisms away from Anosov [7] and its higher-dimensional analogue [8]. Finally, we acknowledge results on the continuity or st”
Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents (2008, background): “Indeed, [6], [7] give a necessary and sufficient condition for a cocycle F over an arbitrary transformation to be a point of continuity of F 7→ λi(F, μ) relative to the C0 = C0,0 topology: the Oseledets splitting must be either dominated or else trivial, over almost every orbit.”
The Lyapunov exponents of generic volume-preserving and symplectic maps (2005, background): “The following notion, introduced in [4], is crucial to the proofs of Theorems 1 through 4.”
GÉNÉRICITÉ D'EXPOSANTS DE LYAPUNOV NON-NULS POUR DES PRODUITS DÉTERMINISTES DE MATRICES GENERICITY OF NON-ZERO LYAPUNOV EXPONENTS FOR DETERMINISTIC PRODUCTS OF MATRICES (2003, background): “Ces théorèmes de Bochi montrent que les résultats sur les produits aléato peuvent pas s’adapter au cas de produits déterministes sans hypothèses supplém Nos conditions portent, d’une part sur la dynamique déterministe qui dirige le co que nous supposerons uniformément hyperbolique, et d’autre part s”
Continuity of Lyapunov exponents for random two-dimensional matrices (2016, methodology): “1 is an adaptation of ideas of Knill [21] and Bochi [4, 5].”
Chapter 2 – Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics (2006, unknown): “6 (Bochi [35]).”