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These lectures will address the dynamics of vector fields or diffeomorphisms of compact manifolds. For the study of generic properties or for the construction of examples, it is often useful to be able to perturb a system. This generally leads to delicate problems: a local modification of the dynamics may cause a radical change in the behavior of the orbits. For the $C^1$-topology, various techniques have been developed which allow to perturb while controlling the dynamics: closing and connection of orbits, perturbation of the tangent dynamics... We derive various applications to the description of $C^1$-generic diffeomorphisms.
These lectures will address the dynamics of vector fields or diffeomorphisms of compact manifolds. For the study of generic properties or for the construction of examples, it is often useful to be able to perturb a system. This generally leads to delicate problems: a local modification of the dynamics may cause a radical change in the behavior of the orbits. For the $C^1$-topology, various techniques have been developed which allow to perturb ...
37C05 ; 37C29 ; 37Dxx
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Smooth parametrization consists in a subdivision of mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the talk is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently separated domains: Smooth Dynamics, Diophantine Geometry, and Approximation Theory. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which I’ll try to stress. Sometimes this similarity can be easily explained, sometimes the reasons remain somewhat obscure, and it motivates some natural questions discussed in the talk. I plan to present also some new results, connecting smooth parametrization with “Remez-type” (or “Norming”) inequalities for polynomials restricted to analytic varieties.
Smooth parametrization consists in a subdivision of mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the talk is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently separated domains: Smooth Dynamics, Diophantine Geometry, and Approximation ...
37C05 ; 11Gxx ; 41A46
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Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers.
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...
37C05 ; 37C40 ; 37D25
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Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers.
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...
37C05 ; 37C40 ; 37D25
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Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers.
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...
37C05 ; 37C40 ; 37D25
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- 105 p.
ISBN 978-0-8218-1943-2
SMF/AMS texts and monographs , 0004
Localisation : Collection 1er étage
anneau # Tore # difféomorphisme # système dynamique dérivable # application
37E30 ; 37C05 ; 37-02
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- 157 p.
ISBN 978-0-19-508545-7
Oxford engineering science series , 0050
Localisation : Ouvrage RdC (EAST)
géométrie # système dynamique discret # point fixe hyperbolique # géométrie symplectique # mesure invariante # exercice # théorie des indexes # index de Conley
37-02 ; 37B30 ; 37B35 ; 37C05
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- 311 p.
ISBN 978-0-8218-3721-4
Fields institute communications , 0048
Localisation : Collection 1er étage
équation différentielle # système dynamique différentiel # dynamique # comportement asymptotique des solutions # problème de valeures limites # application lisse # flux monotone # perturbations variétés de poisson # dynamique des populations # épidémiologie
34-06 ; 37-06 ; 92-06 ; 34C20 ; 34K05 ; 35B40 ; 35J55 ; 37C05 ; 37K55 ; 53D17 ; 92D25 ; 92D30
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- vii; 164 p.
ISBN 978-2-85629-764-3
Astérisque , 0354
Localisation : Périodique 1er étage
Difféomorphisme # généricité # lemme de fermeture # lemme de connection # classe homocline # pseudo-orbite # dynamique hyperbolique # tangence homocline # cycle hétérodimensionnel # hyperbolicité partielle # décomposition dominée # pistage # variété invariante # modèle central # centralisateur
37C05 ; 37C20 ; 37C25 ; 37C29 ; 37C50 ; 37C70 ; 37C75 ; 37D05 ; 37D10 ; 37D15 ; 37D20 ; 37D25 ; 37D30
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- 271 p.
ISBN 978-3-540-66299-0
Lecture notes in mathematics , 1706
Localisation : Collection 1er étage
système dynamique # Shadowing # équation différentielle # EDP # système hamiltonien # système hyperbolique # orbite # ensemble invariant # difféomorphisme # stabilité structurelle # stabilité # application numérique # exponentielle de Lyapunov
37C10 ; 37C50 ; 37C05 ; 37C20 ; 37D05 ; 37L30
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- 244 p.
ISBN
Itogi Nauki i Tekhniki
Localisation : Fonds Russe réserve
équation différentielle # points singulier # cycle limite # système dynamique lisse # système de Morse-Smale # flux des surfaces
34-01 ; 34C05 ; 34C20 ; 34C23 ; 37C05 ; 37D15 ; 34A12 ; 34A07
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