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Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich

Raphaël Krikorian (2003/2004)

Séminaire Bourbaki

Étant donnée une fonction régulière de moyenne nulle sur le tore de dimension 2 , il est facile de voir que ses intégrales ergodiques au-dessus d’un flot de translation “générique”sont bornées. Il y a une dizaine d’années, A. Zorich a observé numériquement une croissance en puissance du temps de ces intégrales ergodiques au-dessus de flots d’hamiltoniens (non-exacts) “génériques”sur des surfaces de genre supérieur ou égal à 2 , et Kontsevich et Zorich ont proposé une explication (conjecturelle) de...

Diffeomorphisms with weak shadowing

Kazuhiro Sakai (2001)

Fundamenta Mathematicae

The weak shadowing property is really weaker than the shadowing property. It is proved that every element of the C¹ interior of the set of all diffeomorphisms on a C ∞ closed surface having the weak shadowing property satisfies Axiom A and the no-cycle condition (this result does not generalize to higher dimensions), and that the non-wandering set of a diffeomorphism f belonging to the C¹ interior is finite if and only if f is Morse-Smale.

Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation

Giuseppe Da Prato, Arnaud Debussche (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup P t ⁢ φ does exist for any bounded φ ; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

For the real quadratic map P a ( x ) = x 2 + a and a given ϵ > 0 a point x has good expansion properties if any interval containing x also contains a neighborhood  J of x with P a n | J univalent, with bounded distortion and B ( 0 , ϵ ) ⊆ P a n ( J ) for some n ∈ ℕ . The ϵ -weakly expanding set is the set of points which do not have good expansion properties. Let α denote the negative fixed point and M the first return time of the critical orbit to [ α , - α ] . We show there is a set ℛ of parameters with positive Lebesgue measure for which the Hausdorff dimension of...

Dimensions des spirales

Yves Dupain, Michel Mendès France, Claude Tricot (1983)

Bulletin de la Société Mathématique de France

Distributional chaos on tree maps: the star case

Jose S. Cánovas (2001)

Commentationes Mathematicae Universitatis Carolinae

Let 𝕏 = { z ∈ ℂ : z n ∈ [ 0 , 1 ] } , n ∈ ℕ , and let f : 𝕏 → 𝕏 be a continuous map having the branching point fixed. We prove that f is distributionally chaotic iff the topological entropy of f is positive.

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