A note on Hölder regularity of invariant distributions for horocycle flows

Cosentino, Salvatore

doi:10.1088/0951-7715/18/6/015

Cited by 7 publications

(14 citation statements)

References 14 publications

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“…For all s > 3/2, the Hilbert space B−s V (SM ) is spanned by the system of finitely-additive measures { β± µ |µ ∈ Spec( ) ∩ R + }. The duality theorem (Theorem 1.7) also leads to a direct bijective correspondence between the lift of the finitely-additive measures β± µ to P SL(2, R) (denoted below by the same symbol) and the Γ-invariant conformal distributions on the boundary of the Poincaré disk studied by S. Cosentino in [9]. Theorem 1.9.…”

Section: Is Not a Coboundary If And Only If The Functionmentioning

confidence: 91%

“…. It was later proved by S. Cosentino in [9] that D ± µ are in fact Hölder of the same orders (that is, they can be written as first derivatives of Hölder continuous functions of exponent 1 ∓ Re ν)/2, except for the distribution D − 1/4 which can be written as a first derivative of a Hölder continuous function of any exponent α < 1/2).…”

Section: Definitions and Notationmentioning

confidence: 99%

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Limit Theorems for Horocycle Flows

Bufetov,

Forni

2011

Preprint

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The main results of this paper are limit theorems for horocycle flows on compact surfaces of constant negative curvature.One of the main objects of the paper is a special family of horocycle-invariant finitely additive Hölder measures on rectifiable arcs. An asymptotic formula for ergodic integrals for horocycle flows is obtained in terms of the finitely-additive measures, and limit theorems follow as a corollary of the asymptotic formula.The objects and results of this paper are similar to those in [15], [16], [4] and [5] for translation flows on flat surfaces. The arguments are based on the classification of invariant distributions for horocycle flows established in [12]. CONTENTS ALEXANDER BUFETOV AND GIOVANNI FORNI 4.4. Proof of the correspondence (Theorem 1.9). 40 5. Proofs of Technical Lemmas. 40 5.1. Outline of the section. 40 5.2. Estimates on coboundaries. 40 5.3. Inner products of cocycles. 44 5.4. Rotationally symmetric measures. 47 References 51

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Section: Is Not a Coboundary If And Only If The Functionmentioning

confidence: 91%

Section: Definitions and Notationmentioning

confidence: 99%

Limit Theorems for Horocycle Flows

Bufetov,

Forni

2011

Preprint

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“…More precisely, one applies (E ± ) n (2.4) and multiplies by the normalizing factor β 2irj ,n = 1 (2irj +1±2n)···(2irj +1±2) . The regularity of these distributions was recently studied in [8,9].…”

Section: Patterson-sullivan Distributionsmentioning

confidence: 99%

“…Otal's proof also shows that the Hölder norm is bounded by a power of r j . Related results can be found in [5,[7][8][9]25].…”

Section: Theorem 32 ([27] Proposition 4) Suppose That S = 1/2 + Ir mentioning

confidence: 99%

Patterson–Sullivan Distributions and Quantum Ergodicity

Anantharaman

Zelditch

2007

Ann. Henri Poincaré

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Abstract. This article gives relations between two types of phase space distributions associated to eigenfunctions φir j of the Laplacian on a compact hyperbolic surface XΓ:, which arise in quantum chaos. They are invariant under the wave group.• Patterson-Sullivan distributions P Sir j , which are the residues of the We prove that these distributions (when suitably normalized) are asymptotically equal as rj → ∞. We also give exact relations between them. This correspondence gives a new relation between classical and quantum dynamics on a hyperbolic surface, and consequently a formulation of quantum ergodicity in terms of classical ergodic theory.

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“…15 By a slight abuse of notations we will still call L F such an extension. 16 The conditions on p, q are not optimal. The lack of optimality begin due to the fact that we require p ∈ N. See [6], and reference therein, for different approaches that remove such a constraint.…”

Section: )mentioning

confidence: 99%

Parabolic dynamics and anisotropic Banach spaces

Giulietti

Liverani

2019

J. Eur. Math. Soc.

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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].

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A note on Hölder regularity of invariant distributions for horocycle flows

Cited by 7 publications

References 14 publications

Limit Theorems for Horocycle Flows

Limit Theorems for Horocycle Flows

Patterson–Sullivan Distributions and Quantum Ergodicity

Parabolic dynamics and anisotropic Banach spaces

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