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# Documents  53D25 | enregistrements trouvés : 15

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## Persistence modules and Hamiltonian diffeomorphisms - Part 1 Polterovich, Leonid | CIRM H

Post-edited

Research schools;Exposés de recherche;Dynamical Systems and Ordinary Differential Equations;Geometry

Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

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## A microlocal toolbox for hyperbolic dynamics Dyatlov, Semyon | CIRM H

Post-edited

Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Geometry

I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between shift to higher frequencies and escaping in the physical space. I will show meromorphic continuation of the resolvent of $X$; the poles, known as Pollicott-Ruelle resonances, describe exponential decay of correlations. As an application, I will prove that the Ruelle zeta function continues meromorphically for flows on non-compact manifolds (the compact case, known as Smale's conjecture, was recently settled by Giulietti-Liverani- Pollicott and a simple microlocal proof was given by Zworski and the speaker). Joint work with Colin Guillarmou. I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between ...

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## Ecole de théorie ergodiqueMarseille # avril 2006 Lacroix, Yves ; Liardet, Pierre ; Thouvenot, Jean-Paul | Société Mathématique de France 2010

Congrès

- xxii; 266 p.
ISBN 978-2-85629-312-6

Séminaires et congrès , 0020

Localisation : Collection 1er étage

bêta-numération # application premier retour # applications d'intervalles # applications fer à cheval # applications monotones par morceaux # attracteurs # auto-couplage # automate cellulaire # cobord # convolutions de Bernoulli # décalages markoviens fortement positivement récurrents # développements glouton et paresseux # diagramme de Markov # dimension de Hausdorff # dynamique symbolique # dynamiques directionnelles # échelles de numération # entropie # feuilletage linéaire # flot géodésique # géométrie fractale # invariants par tricotage # mélange faible # mesure d'Erdös # mesure invariante # mesure invariante absolument continue # mesures de Gibbs et faiblement gibbsiennes # mesures d'entropie maximale # métrique euclidienne # nombre d'or # odomètre # orbites périodiques # partition markovienne # pistage # principe variationnel # problème de rigidité # rang faible # section transverse # sous-décalage de Toeplitz # surface plate # surface pointée # système dynamique # systèmes dynamiques en topologie et en combinatoire # systèmes dynamiques minimaux # systèmes dynamiques symboliques # théorie des suites de tricotage # théorie ergodique # transformation de rang un # zêta fonction de Artin-Mazur bêta-numération # application premier retour # applications d'intervalles # applications fer à cheval # applications monotones par morceaux # attracteurs # auto-couplage # automate cellulaire # cobord # convolutions de Bernoulli # décalages markoviens fortement positivement récurrents # développements glouton et paresseux # diagramme de Markov # dimension de Hausdorff # dynamique symbolique # dynamiques directionnelles # échelles de numération # ...

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## Geodesic flows on closed Riemann manifolds with negative curvature Anosov, D. V. | American Mathematical Society 1969

Congrès

- 235 p.
ISBN 978-0-8218-1890-9

Proceedings of the Steklov institute of mathematics , 0090

Localisation : Collection 1er étage

géométrie différentielle # flux géodésique # variété riemannienne # courbure négative

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## $S$-adic sequences: a bridge between dynamics, arithmetic, and geometry Thuswaldner, Jörg | CIRM H

Multi angle

Research schools;Dynamical Systems and Ordinary Differential Equations;Geometry;Number Theory

Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity.
It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy’s program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization.
Nevertheless, recently Berthé, Steiner, and Thuswaldner (2016) made some further progress on Rauzy’s program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982).
Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincaré section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin’s idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well.
It is the aim of my series of lectures to review the above results.
Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword ...

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## Persistence modules and Hamiltonian diffeomorphisms - Part 4 Polterovich, Leonid | CIRM H

Multi angle

Research schools;Dynamical Systems and Ordinary Differential Equations;Geometry

Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

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## Persistence modules and Hamiltonian diffeomorphisms - Part 3 Polterovich, Leonid | CIRM H

Multi angle

Research schools;Exposés de recherche;Dynamical Systems and Ordinary Differential Equations;Geometry

Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

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## Persistence modules and Hamiltonian diffeomorphisms - Part 2 Polterovich, Leonid | CIRM H

Multi angle

Research schools;Dynamical Systems and Ordinary Differential Equations;Geometry

Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

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## Spectrum of geodesic flow on negatively curved manifold Tsujii, Masato | CIRM H

Multi angle

Research talks;Dynamical Systems and Ordinary Differential Equations;Mathematical Physics

We consider the one-parameter families of transfer operators for geodesic flows on negatively curved manifolds. We show that the spectra of the generators have some "band structure" parallel to the imaginary axis. As a special case of "semi-classical" transfer operator, we see that the eigenvalues concentrate around the imaginary axis with some gap on the both sides.

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## Eigenfunctions of the Laplacien on a Riemannian manifold Zelditch, Steve | American Mathematical Society;National science foundation 2017

Ouvrage

- xiv; 394 p.
ISBN 978-1-4704-1037-7

CBMS regional conference series in mathematics , 0125

Localisation : Collection 1er étage

analyse globale # fonction propre # Laplacien # variété Riemannienne # opérateur intégral de Fourier # équation d'onde # flot géodésique

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## Free loop spaces in geometry and topology:including the monograph "Symplectic cohomology and Viterbo's theorem" by Mohammed Abouzaid Latschev, Janko ; Oancea, Alexandru | European Mathematical Society 2015

Ouvrage

- vi; 494 p.
ISBN 978-3-03719-153-8

IRMA lectures in mathematics and theoretical physics , 0024

Localisation : Ouvrage RdC (FREE)

espace de lacets # géométrie symplectique # topologie symplectique et de contact # théorie de Morse # homologie cyclique # homologie de Hochschild # homotopie rationnelle # modèle minimal # plongement lagrangien # courbe pseudo-holomorphe # espace de modules # opérateur de Cauchy-Riemann # théorie de Floer # cohomologie symplectique

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## Séminaire Bourbaki. Vol. 2014/2015: exposés 1089-1103 | Société Mathématique de France 2016

Ouvrage

- x; 497 p.
ISBN 978-2-85629-836-7

Astérisque , 0380

Localisation : Périodique 1er étage

combinatoire # théorie des catégories # théorie des topos supérieurs # théorie de la mesure géométrique # équation aux dérivées partielles # théorie spectrale # géométrie différentielle # théorie ergodique # théorie géométrique des groupes # géométrie algébrique # représentation galoisienne # point rationnel

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## Equilibrium states in negative curvature Paulin, Frédéric ; Pollicott, Mark ; Schapira, Barbara | Société Mathématique de France 2015

Ouvrage

- viii; 281 p.
ISBN 978-2-85629-818-3

Astérisque , 0373

Localisation : Périodique 1er étage

flot géodésique # courbure négative # état de Gibbs # période # dénombrement d'orbites # densité de Patterson # pression # principe variationnel # foliation instable et forte

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## Integrable hamiltonian systems :geometry, topology, classification Bolsinov, Alexey V. ; Fomenko, Anatoly T. | Chapman & Hall/CRC 2004

Ouvrage

- 730 p.
ISBN 978-0-415-29805-6

Localisation : ouvrage RdC (BOLS)

système intégrable # système hamiltonien # flot géodésique # classification de Liouville

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## Dynamics and entropies of Hilbert metrics Crampon, Mickaël | Institut de Recherche Mathématique Avancée Strasbourg (IRMA) 2011

Thèse

- xxii; 113 p.

Localisation : Ouvrage RdC (CRAM)

géométrie de Hilbert # système dynamique hyperbolique # flot géodésique # entropie # exposant de Lyapunov

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