Banach spaces adapted to Anosov systems

By S. Gouëzel, C. Liverani. Ergodic Theory and Dynamical Systems, 2004

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We study the spectral properties of the Ruelle–Perron–Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the ${\mathcal C}^\infty$ case, the essential spectral radius is arbitrarily small, which yields a description of the correlations with arbitrary precision. Moreover, we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the Sinai–Ruelle–Bowen measure, the variance for the central limit theorem, the rates of decay for smooth observable, etc.).

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/43ddedeae975b7d3e14ab48a26cd9d518c192909

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Citations in the corpus

Dynamical determinants and spectrum for hyperbolic diffeomorphisms (2006, methodology): “Since the spaces of functions in this paper and [7], as well as those in [ 15 ], are completions of the space of Cr−1 functions and embedded in the space of distributions, the lemma above tells that the part of spectrum of L outside of the essential spectral radius does not depend on those choices o”
Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow (2010, background): “This kind of arguments has been developed in recent years for Anosov maps or ﬂows in compact manifolds and has proved very fruitful (see among others [Liv04, GL06, BT07, BL07]).”
Quenched CLT for random toral automorphism (2007, background): “To study such operators it is necessary to introduce appropriate Banach spaces (see [13, 5, 12]).”
Good Banach spaces for piecewise hyperbolic maps via interpolation (2007, methodology): “dynamics, good spaces of distributions were only introduced a few years ago by Gou¨ezel–Liverani and Baladi–Tsujii [ 20 , 5, 21, 6], following the pioneering work of Blank–Keller–Liverani [8].”
Decay of correlations in suspension semi-flows of angle-multiplying maps (2005, background): “For the cases of discrete dynamical systems such as iterations of expanding maps and Anosov diffeomorphisms, exponential decay of correlations for Hölder observables was already known in the early stage of the study [6, 16, 17] and nowadays we have a fairly good understanding of the speed of decay a”
ON ULAM APPROXIMATION OF THE ISOLATED SPECTRUM AND EIGENFUNCTIONS OF HYPERBOLIC MAPS (2006, background): “Quasi-compactness of the Perron-Frobenius operator has been proven in a variety of settings [23, 16, 22, 5, 14, 2].”
Smooth Anosov flows: Correlation spectra and stability (2007, methodology): “Remark 6.1. Note that the perturbation theory in [18] and [ 12 ] allows to investigate, by similar arguments, also the behaviour of the other eigenvalues of X , with the related eigenspaces, outside the essential spectrum.”
Anosov flows and dynamical zeta functions (2012, methodology): “The above spaces are the natural extensions of the spaces B in [19] to the case of `-forms.”
Anisotropic Sobolev spaces and dynamical transfer operators: C (2004, background): “ǫ (for ǫ>0) will be treated in a forthcoming work [4], using spaces due to Alinhac [1]. We hope that the (general) Cα foliation case will be amenable to the present approach. Gou¨ezel and Liverani [10] have independently obtained non trivial, but weaker, bounds for the essential spectral radius o”