Complete spectral data for analytic Anosov maps of the torus

Slipantschuk, Julia; Bandtlow, Oscar F.; Just, Wolfram

doi:10.1088/1361-6544/aa700f

Cited by 25 publications

(46 citation statements)

References 47 publications

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“…In order to demonstrate this phenomenon, we resort to analytical solutions of two-dimensional hyperbolic diffeomorphisms which allow for fractal invariant measures if the Jacobian is not constant. The presence of contracting and expanding directions requires using more involved function spaces, that is, a particular class of anisotropic Hilbert spaces, for which rigorous statements on spectral data of evolution operators are possible (see, for example, [29] for technical details). We illustrate our point by considering an analytic deformation of the cat map given by (ϕ 1 , ϕ 2 ) → (ϕ ′ 1 , ϕ ′ 2 ) with ϕ ′ 1 =2ϕ 1 + ϕ 2 + 2arctan |µ| sin(ϕ 1 + ϕ 2 − α) 1 − |µ| cos(ϕ 1 + ϕ 2 − α) ,…”

Section: Discussionmentioning

confidence: 99%

Dynamic mode decomposition for analytic maps

Slipantschuk

Bandtlow

Just

2020

Communications in Nonlinear Science and Numerical Simulation

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Extended dynamic mode decomposition (EDMD) provides a class of algorithms to identify patterns and effective degrees of freedom in complex dynamical systems. We show that the modes identified by EDMD correspond to those of compact Perron-Frobenius and Koopman operators defined on suitable Hardy-Hilbert spaces when the method is applied to classes of analytic maps. Our findings elucidate the interpretation of the spectra obtained by EDMD for complex dynamical systems. We illustrate our results by numerical simulations for analytic maps.

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Section: Discussionmentioning

confidence: 99%

Dynamic mode decomposition for analytic maps

Slipantschuk

Bandtlow

Just

2020

Communications in Nonlinear Science and Numerical Simulation

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“…45 Though some hope is given by the construction of generic examples, for the operator associated to the SRB measure, with spectrum different from {0, 1} by Alexander Adam [1]. See also the more recent results in [8] based on special examples described in [54] for which the spectrum can be explicitly computed.…”

Section: A One-parameter Family Of Examplesmentioning

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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“…The existence of such function spaces "adapted to the hyperbolic dynamics" follow in the analytic category from the work of Faure-Roy [9]. More recently, these function spaces have been used to study the Ruelle spectrum of Anosov maps by Adam [1], and also Bandtlow-Just-Slipantschuk [16]. We will provide more details for the construction of these spaces later on but roughly speaking, they are designed in Fourier coordinates to impose analyticity in the stable direction (exponential decay of Fourier coefficients) while irregularity is allowed in the unstable direction (exponential growth at most of Fourier modes).…”

Section: Function Space and Reduction To A Spectral Problemmentioning

confidence: 99%

On the rate of mixing of circle extensions of Anosov maps

Naud

2018

J. Spectr. Theory

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Let A : T 2 → T 2 be an Anosov diffeomorphism. Circle extensions A are a rich family of non-uniformly hyperbolic diffeomorphisms living on T 2 × S 1 for which the rate of mixing is conjectured to be generically exponential. In this paper, we investigate the possible rates of exponential mixing by exhibiting some explicit lower bounds on the decay rate by Fourier analytic and probabilistic techniques. The rates obtained are related to the topological pressure of two times the unstable jacobian.

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Complete spectral data for analytic Anosov maps of the torus

Cited by 25 publications

References 47 publications

Dynamic mode decomposition for analytic maps

Dynamic mode decomposition for analytic maps

Parabolic dynamics and anisotropic Banach spaces

On the rate of mixing of circle extensions of Anosov maps

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