Ruelle?Perron?Frobenius spectrum for Anosov maps

By M. Blank, Gerhard Keller, C. Liverani. 2001

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We extend a number of results from one-dimensional dynamics based on spectral properties of the Ruelle–Perron–Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows us to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasi-compact. (Information on the existence of a Sinai–Ruelle–Bowen measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated with smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows us to obtain easily very strong spectral stability results, which, in turn, imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam-type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite-dimensional problem.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/14324f6bda47e17cc8f52d2484924a42d816bbf0

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Citations in the corpus

On contact Anosov flows (2004, methodology): “In fact, employing the strategy used in [ 2 ], section 3.6, and refining”
Smooth Anosov flows: Correlation spectra and stability (2007, background): “Starting with [28, 5 ] it has become clear that it is possible”
Banach spaces for piecewise cone hyperbolic maps (2009, unknown): “The following spectral gap result of 1 Blank–Keller–Liverani [BKL02] appeared in 2002: Theorem 1.1.”
Almost-invariant sets and invariant manifolds — Connecting probabilistic and geometric descriptions of coherent structures in flows (2009, background): “Convergence of the peripheral spectrum and eigenprojections of the transition matrix (7) to the true transfer operator as the partition is refined has been considered in a variety of autonomous settings [39, 40, 41].”
Chapter 2 – Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics (2006, background): “choice of Banach spaces due to Gouëzel & Liverani [100] and Blank, Keller & Liverani [34] which avoids symbolic dynamics, and is thus better suited to the study of such issues.”
Anisotropic Sobolev spaces and dynamical transfer operators: C (2004, methodology): “ral radius of M, on a diﬀerent Banach space, in this general case. We end this introduction with three open problems: Remark 1.1 (Links with SRBmeasures). With the techniques ofBlank– Keller–Liverani [6], one should obtain that the spectral radius of L on each Wp,q,t(X) is one, that 1 is is a semi-s”
Quenched CLT for random toral automorphism (2007, background): “Instead of appealing to the general theory developed in [6, 4, 12, 5, 13] we will take advantage of the simplicity of the present setting and introduce explicitly a particularly simple version of such a theory.”
On Contact Anosov Flows (2003, methodology): “sov Contact ﬂows exhibit exponential decay of correlations. This is exactly what it is proved in the present paper (Theorem 2.4). Toobtain such a resultI built on Dolgopyat’sworkand on the results in [2] where it is introduced a functional space over which the Perron-Frobenius operator can be studie”
Anosov flows and dynamical zeta functions (2012, background): “The basic idea, going back to [12], is to consider “appropriate objects” that are “smooth” in the unstable direction while being “distributions” in the stable direction.”
Banach spaces adapted to Anosov systems (2004, background): “Indeed, the ideas in [ 7 ]h ave already been applied with success to some partially hyperbolic situations (flows) [19 ]a nd we expect them to be applicable to the study of dynamical zeta functions.”
Dynamical determinants and spectrum for hyperbolic diffeomorphisms (2006, background): “ diﬃcult to go beyond the results on dynamical zeta functions by Parry and Pollicott (see [21]). (See Section 2 for a discussion about dynamical zeta functions ζT,g(z).) Recently, in a pioneering work[9], Blank, Keller and Liverani introduced scales of Banach spaces of distributions on the manifold ”