Parabolic dynamics and anisotropic Banach spaces

Abstract

We investigate the relation between the distributions appearing in the study of ergodic averages of parabolic flows (e.g. in the work of Flaminio-Forni) and the ones appearing in the study of the statistical properties of hyperbolic dynamical systems (i.e. the eigendistributions of the transfer operator). In order to avoid, as much as possible, technical issues that would cloud the basic idea, we limit ourselves to a simple flow on the torus. Our main result is that, roughly, the growth of ergodic averages (and the characterization of coboundary regularity) of a parabolic flows is controlled by the eigenvalues of a suitable transfer operator associated to the renormalizing dynamics. The conceptual connection that we illustrate is expected to hold in considerable generality.2000 Mathematics Subject Classification. 37A25, 37A30, 37C10, 37C40, 37D40. Key words and phrases. Horocycle flows, quantitative equidistribution, quantitative mixing, spectral theory, transfer operator. L.C. gladly thanks Giovanni Forni for several discussions through the years. Such discussions began in the very far past effectively starting the present work and lasted all along, in particular the last version of Lemma 5.11 owns to Forni ideas. P.G. thanks L. Flaminio for explaining parts of his work and his hospitality at the university of Lille. Also we would like to thank Viviane Baladi, Alexander Bufetov, Oliver Butterley, Ciro Ciliberto, Livio Flaminio, François Ledrappier, Frédéric Naud, René Schoof and Corinna Ulcigrai for several remarks and suggestions that helped to considerably improve the presentation. We thank the anonymous referee for his hard work and thorough suggestions which forced us to substantially improve the paper. the IHES for the kind hospitality during the revision of the paper. 1 But see, e.g., [49,50,43,41] for earlier related results.2 Typical examples are circle rotations [52, 53], interval exchange maps via Teichmuller theory [55,28], horocycle flow [26,21].

Ruelle spectrum of linear pseudo-Anosov maps

“…We will obtain a complete description of the Ruelle spectrum of linear pseudo-Anosov map. Then, using the philosophy of Giulietti-Liverani [GL14] that Ruelle resonances contain information on the translation flow along the stable manifold on the map, we will discuss consequences of these results on the vertical translation flow in translation surfaces supporting a pseudo-Anosov map. We will in particular obtain complete results on the set of distributions which are invariant under the vertical flow, and on smooth solutions to the cohomological equation, recovering in this case results due to Forni on generic translation surfaces [For97,For02,For07].…”

The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces

“…where Θ is a smooth compactly supported function. A method to overcome this difficulty was recently developed in [42], where the authors decompose the observable (3.7) into a infinite sum of observables satisfying (3.8). It turns out that the resonances λ k in (1.5) leading to α k > 0 correspond to deviations of ergodic sums while the resonances leading to α k ≤ 0 are responsible for obstructions for having smooth solutions to the cohomological equation.…”

Generic non-trivial resonances for Anosov diffeomorphisms

“…We put c := c 1 and M in Definitions 2.2 and 2.4, giving a linear map A M,c1 and a Hilbert space H AM,c 1 . Recalling (6), and assuming that K T :…”