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Journal of Modern Dynamics

January 2012 , Volume 6 , Issue 1

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Équidistribution, comptage et approximation par irrationnels quadratiques
Jouni Parkkonen and Frédéric Paulin
2012, 6(1): 1-40 doi: 10.3934/jmd.2012.6.1 +[Abstract](590) +[PDF](756.1KB)
Abstract:
Soit $M$ une variété hyperbolique de volume fini, nous montrons que les hypersurfaces équidistantes à une sous-variété $C$ de volume fini totalement géodésique s'équidistribuent dans $M$. Nous donnons une asymptotique précise du nombre de segments géodésiques de longueur au plus $t$, perpendiculaires communs à $C$ et au bord d'un voisinage cuspidal de $M$. Nous en déduisons des résultats sur le comptage d'irrationnels quadratiques sur $\mathbb{Q}$ ou sur une extension quadratique imaginaire de $\mathbb{Q}$, dans des orbites données des sous-groupes de congruence des groupes modulaires.

Let $M$ be a finite volume hyperbolic manifold. We show the equidistribution in $M$ of the equidistant hypersurfaces to a finite volume totally geodesic submanifold $C$. We prove a precise asymptotic formula on the number of geodesic arcs of lengths at most $t$, that are perpendicular to $C$ and to the boundary of a cuspidal neighbourhood of $M$. We deduce from it counting results of quadratic irrationals over $\mathbb{Q}$ or over imaginary quadratic extensions of $\mathbb{Q}$, in given orbits of congruence subgroups of the modular groups.
On primes and period growth for Hamiltonian diffeomorphisms
Ely Kerman
2012, 6(1): 41-58 doi: 10.3934/jmd.2012.6.41 +[Abstract](575) +[PDF](254.2KB)
Abstract:
Here we use Vinogradov's prime distribution theorem and a multidimensional generalization due to Harman to strengthen some recent results from [12] and [13] concerning the periodic points of Hamiltonian diffeomorphisms. In particular we establish resonance relations for the mean indices of the fixed points of Hamiltonian diffeomorphisms which do not have periodic points with arbitrarily large periods in $\mathbb{P}^2$, the set of natural numbers greater than one which have at most two prime factors when counted with multiplicity. As an application of these results we extend the methods of [2] to partially recover, using only symplectic tools, a theorem on the periodic points of Hamiltonian diffeomorphisms of the sphere by Franks and Handel from [10].
Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition
Claire Chavaudret and Stefano Marmi
2012, 6(1): 59-78 doi: 10.3934/jmd.2012.6.59 +[Abstract](697) +[PDF](252.7KB)
Abstract:
The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasiperiodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize previous works by L.H. Eliasson which deal with the diophantine case so as to implement a Brjuno-Rüssmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Pöschel-Rüssmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles.
Hölder foliations, revisited
Charles Pugh, Michael Shub and Amie Wilkinson
2012, 6(1): 79-120 doi: 10.3934/jmd.2012.6.79 +[Abstract](844) +[PDF](1522.4KB)
Abstract:
We investigate transverse Hölder regularity of some canonical leaf conjugacies in normally hyperbolic dynamical systems and transverse Hölder regularity of some invariant foliations. Our results validate claims made elsewhere in the literature.
Genericity of nonuniform hyperbolicity in dimension 3
Jana Rodriguez Hertz
2012, 6(1): 121-138 doi: 10.3934/jmd.2012.6.121 +[Abstract](749) +[PDF](456.3KB)
Abstract:
For a generic conservative diffeomorphism of a closed connected 3-manifold $M$, the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is nonuniformly hyperbolic and ergodic.
    This is the 3-dimensional version of the well-known result by Mañé-Bochi [14, 4], stating that a generic conservative surface diffeomorphism is either Anosov or all Lyapunov exponents vanish almost everywhere. This result was inspired by and answers in the positive in dimension 3 a conjecture by Avila-Bochi [2].

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