Abstract : ­See Gromov ­ : Recent: “In a Search for a Structure, Part 1: On Entropy”

Abstract : This is a joint work with Masato Tsujii (Kyushu Univ.).
We consider any smooth symplectic Anosov diffeomorphism f on a compact symplectic manifold M. This is considered as a standard model of "chaotic dynamics".
Based on the work of Durhuus-Jonsson and Benedetti-Ziegler, we revisit the question of the number of triangulations of the 3-ball. We introduce a notion of nucleus (a triangulation of the 3-ball without internal nodes, and with each internal face having at most 1 external edge). We show that every triangulation can be built from trees of nuclei. This leads to a new reformulation of Gromov's question: We show that if the number of rooted nuclei with t tetrahedra is exponentially bounded in t, then the number of rooted triangulations with t tetrahedra is also exponentially bounded. This is joint work with Pierre Collet and Maher Younan.­

Résumé : This is a joint work with Masato Tsujii (Kyushu Univ.).
We consider any smooth symplectic Anosov diffeomorphism f on a compact symplectic manifold M. This is considered as a standard model of "chaotic dynamics".
Following Kostant, Souriau, Kirillov (70') we consider the "prequantum bundle" P, which is the U(1)-principal bundle over M, whose curvature is the symplectic form, and the "prequantum map" which is the map f lifted on P in the natural way. This lifted map F is partially hyperbolic and mixing. The later means that a smooth probability measure which evolves under F (weakly-) converges exponentially towards the uniform equilibrium measure on P. We study the fluctuations around this equilibrium following the spectral approach of David Ruelle: we show that the discrete spectrum of "Ruelle resonances" of this dynamics (i.e. the spectrum of the pull back operator by F in appropriate Sobolev spaces and restricted to the Fourier mode N along the fibers) has a particular structure: for each and large enough N, there is an external annulus containing a finite number of eigenvalues separated from the rest of the internal spectrum by a spectral gap. This means that the long time classical correlation functions are described  by a finite rank operator, up to exponential decaying errors. We show that this finite rank operator has the properties of a "quantum map" i.e. a "quantization of f" with the Planck constant h=1/N: it satisfies the Gutzwiller Trace formula, the exact Egorov theorem, etc, its rank is given the Weyl formula (more precisely the Atiyah-Singer index formula). In the special case of the linear Arnold cat map on the torus, this quantum map coincides with the Weyl quantization of f. The results are obtained using a semiclassical approach similar to the semiclassical theory of quantum scattering developped by Helffer-Sjöstrand (80').­

Résumé : This talk is about concentration inequalities for a large class of nonuniformly hyperbolic dynamical systems. For dynamical systems modeled by a Young tower with exponential tails, we prove an exponential concentration inequality for all separately Lipschitz observables of n variables. When tails are polynomial, we prove polynomial concentration inequalities. Those inequalities are optimal. We give some applications of such inequalities to specific systems and specific observables.