Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices

Abstract

We propose a geometric sufficient criterium "à la Furstenberg" for the existence of non-zero Lyapunov exponents for certain linear cocycles over hyperbolic transformations: non-existence of probability measures on the fibers invariant under the cocycle and under the holonomies of the stable and unstable foliations of the transformation. This criterium applies to locally constant and to dominated cocycles over hyperbolic sets endowed with an equilibrium state.As a consequence, we get that non-zero exponents exist for an open dense subset of these cocycles, which is also of full Lebesgue measure in parameter space for generic parametrized families of cocycles.This criterium extends to continuous time cocycles obtained by lifting a hyperbolic flow to a projective fiber bundle, tangent to some foliation transverse to the fibers. Again, non-zero Lyapunov exponents are implied by non-existence of transverse measures invariant under the holonomy of the foliation.We apply this last result to a natural geometric context: the geodesic flow tangent to the leaves of a foliation obtained as the suspension of a representation ρ : π 1 (S) → PSL(2, C) of the fundamental group of a hyperbolic compact surface. We prove the existence of non-zero Lyapunov exponents for the corresponding cocycle, for ρ in a dense open subset of all the representations. As a consequence we get that this foliated geodesic flow has a unique Sinai-Ruelle-Bowen measure.

Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents

“…The conclusion of Corollary C was obtained before by Bonatti, GomezMont, Viana [9], under the additional assumptions that the measure is ergodic and the cocycle has a partial hyperbolicity property called domination. Then the set A may be chosen independent of µ.…”

“…Bonatti, GomezMont, Viana [9] obtained a version of Furstenberg's positivity criterion that applies to any cocycle admitting invariant stable and unstable holonomies, and Bonatti, Viana [10] similarly extended the Guivarc'h, Raugi simplicity crite-rion. The condition on the invariant holonomies is satisfied, for instance, if the cocycle is either locally constant or dominated.…”

Extremal Lyapunov exponents: an invariance principle and applications

“…Proof. The claims follow from the same partial hyperbolicity methods (see Hirsch, Pugh, Shub [27]) used before to obtain similar results for linear cocycles [14,16,43], and so we just sketch the main ingredients. Existence (1) and invariance (2) of the family W s follow from a standard application of the graph transform argument [27].…”

“…We take the cocycleF :Ê →Ê to satisfy a normal hyperbolicity property similar to the center bunching condition of Burns, Wilkinson [18] and which was first introduced in [14] in the context of linear cocycles. We say that a Lipschitz smooth cocycleF is dominated if there exist ≥ 1 and θ < 1 such that (6) (…”

“…Assuming the base dynamics (g, µ) is fairly "chaotic" (hyperbolic, possibly in a non-uniform fashion), there is now a good understanding of such issues as the existence of non-zero Lyapunov exponents (see Bonatti, Gomez-Mont, Viana [14,43]) or the simplicity of the Lyapunov spectrum (see Avila, Bonatti, Viana [5,6,16]), in line with the classical theory of random matrices developed by Furstenberg [23], Ledrappier [33], Guivarc'h, Raugi [25], Gol'dsheid, Margulis [24], and other authors. In a nutshell, for generic linear cocycles the Lyapunov exponents are not all zero.…”