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Masse des pointes, temps de retour et enroulements en courbure négative

Nathanaël Enriquez, Jacques Franchi (2002)

Bulletin de la Société Mathématique de France

Soient Γ un groupe discret géométriquement fini d’isométries d’une variété de Hadamard pincée X et 𝒫 une pointe de l’orbifold associé ℳ : = Γ ∖ X . Munissant T 1 ℳ de sa mesure de Patterson-Sullivan m , nous obtenons une estimation asymptotique de la masse d’un petit voisinage horocyclique de 𝒫 , moyennant une hypothèse sur la croissance du sous-groupe parabolique associé à 𝒫 , hypothèse qui est réalisée si X est symétrique de rang 1 . Nous en déduisons une estimation asymptotique du temps de retour du flot géodésique...

Maximal distributional chaos of weighted shift operators on Köthe sequence spaces

Xinxing Wu (2014)

Czechoslovak Mathematical Journal

During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator B w n : λ p ( A ) → λ p ( A ) defined on the Köthe sequence space λ p ( A ) exhibits distributional ϵ -chaos for any 0 < ϵ < diam λ p ( A ) and any n ∈ ℕ is obtained. Under this assumption, the principal measure of B w n is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional ϵ -chaos for any 0 < ϵ < diam λ p ( A ) .

Maximal scrambled sets for simple chaotic functions.

Víctor Jiménez López (1996)

Publicacions Matemàtiques

This paper is a continuation of [1], where a explicit description of the scrambled sets of weakly unimodal functions of type 2∞ was given. Its aim is to show that, for an appropriate non-trivial subset of the above family of functions, this description can be made in a much more effective and informative way.

Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants

Alex Eskin, Howard Masur, Anton Zorich (2003)

Publications Mathématiques de l'IHÉS

A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesics. We give a complete description of...

Multibump solutions for Hamiltonian systems with fast and slow forcing

Vittorio Coti Zelati, Margherita Nolasco (1999)

Bollettino dell'Unione Matematica Italiana

Si dimostra l'esistenza di infinite soluzioni «multi-bump» - e conseguentemente il comportamento caotico - per una classe di sistemi Hamiltoniani del secondo ordine della forma - q ¨ + q = g 1 ⁢ ω ⁢ t + g 2 ⁢ t / ω ⁢ V ′ ⁢ q per ω sufficientemente piccolo. Qui q ∈ R n , g 1 e g 2 sono funzioni strettamente positive e periodiche e V è un potenziale superquadratico (ad esempio V ⁢ q = q 4 ).

Multifractal spectra of Birkhoff averages for a piecewise monotone interval map

Franz Hofbauer (2010)

Fundamenta Mathematicae

We study the entropy spectrum of Birkhoff averages and the dimension spectrum of Lyapunov exponents for piecewise monotone transformations on the interval. In general, these transformations do not have finite Markov partitions and do not satisfy the specification property. We characterize these multifractal spectra in terms of the Legendre transform of a suitably defined pressure function.

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