## ANDERSSON, Martin Stable ergodicity of dominated systems

Reference: DMA-09-01 (February 2009)

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Abstract: We provide a new approach to stable ergodicity of systems with dominated splittings, based on a geometrical analysis of global stable and unstable manifolds of hyperbolic points. Our method suggests that the lack of uniform size of Pesin's local stable and unstable manifolds --- a notorious problem in the theory of non-uniform hyperbolicity --- is often less severe than it appeas to be.

AMS Classification :****
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## Barbosa Gomes, José, Ruggiero, Rafael O. On Finsler surfaces without conjugate points

Reference: DMA-09-02 (March 2009)

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Abstract: We show that \$C^{3}\$ compact Finsler surfaces of genus \$\geq 2\$ without conjugate points, which are either of Landsberg type or k-basic with expansive geodesic flow, are Riemannian. The key point of the proof is to show that first integrals of Finsler geodesic flows without conjugate points in compact surfaces of higher genus are constant. This fact generalizes a previous result by G. Paternain in the case of analytic Finsler compact surfaces of higher genus. The connection between the absence of conjugate points and the Riemannian character of the Finsler metric has some remarkable consequences concerning rigidity.

AMS Classification :****
Keywords :Finsler metric, geodesic flow, Riemannian metric, conjugate points, expansiveness

## BALADI, Viviane, GOU\"{E}ZEL, Sébastien Banach spaces for piecewise cone hyperbolic maps

Reference: DMA-09-03 (August 2009)

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Abstract: We consider piecewise cone hyperbolic systems satisfying a bunching condition and we obtain a bound on the essential spectral radius of the associated weighted transfer operators acting on anisotropic Sobolev spaces. The bunching condition is always satisfied in dimension two, and our results give a unifying treatment of the work of Demers-Liverani \cite{demers_liverani} and our previous work \cite{baladi_gouezel_piecewise}. When hyperbolicity dominates complexity, our bound implies a spectral gap for the transfer operator corresponding to the physical measures.

AMS Classification :****
Keywords :****

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