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Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices. (Genericity of non-zero Lyapunov exponents for deterministic products of matrices). (French) Zbl 1025.37018
Authors’ 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 \(\rho:\pi_1(S)\to \text{PSL} (2,\mathbb{C})\) of the fundamental group of a hyperbolic compact surface. We prove the existence of non-zero Lyapunov exponents for the corresponding cocycle, for \(\rho\) 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.

37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37C20 Generic properties, structural stability of dynamical systems
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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