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Sur un théorème de Dulac

Laurent Stolovitch (1994)

Annales de l'institut Fourier

Nous considérons les champs de vecteurs analytiques de ( ℂ n , 0 ) de partie linéaire diagonale non nulle et dont les valeurs propres λ i ∈ ℂ vérifient des relations de résonances toutes engendrées par une seule relation ( r , λ ) = 0 pour un certain vecteur r ∈ ℕ n non nul. Nous montrons que, dans un système de coordonnées locales holomorphes au voisinages de 0 ∈ ℂ n , de tels champs de vecteurs se “mettent" sous une forme normale partielle, tout en exhibant des variétés invariantes, si l’on fait une hypothèse de petits diviseurs diophantiens....

Symbolic extensions for nonuniformly entropy expanding maps

David Burguet (2010)

Colloquium Mathematicae

A nonuniformly entropy expanding map is any ¹ map defined on a compact manifold whose ergodic measures with positive entropy have only nonnegative Lyapunov exponents. We prove that a r nonuniformly entropy expanding map T with r > 1 has a symbolic extension and we give an explicit upper bound of the symbolic extension entropy in terms of the positive Lyapunov exponents by following the approach of T. Downarowicz and A. Maass [Invent. Math. 176 (2009)].

Symbolic extensions in intermediate smoothness on surfaces

David Burguet (2012)

Annales scientifiques de l'École Normale Supérieure

We prove that 𝒞 r maps with r > 1 on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].

Szegő's first limit theorem in terms of a realization of a continuous-time time-varying systems

Pablo Iglesias, Guoqiang Zang (2001)

International Journal of Applied Mathematics and Computer Science

It is shown that the limit in an abstract version of Szegő's limit theorem can be expressed in terms of the antistable dynamics of the system. When the system dynamics are regular, it is shown that the limit equals the difference between the antistable Lyapunov exponents of the system and those of its inverse. In the general case, the elements of the dichotomy spectrum give lower and upper bounds.

Tame semiflows for piecewise linear vector fields

Daniel Panazzolo (2002)

Annales de l’institut Fourier

Let ℰ be a disjoint decomposition of ℝ n and let X be a vector field on ℝ n , defined to be linear on each cell of the decomposition ℰ . Under some natural assumptions, we show how to associate a semiflow to X and prove that such semiflow belongs to the o-minimal structure ℝ an , exp . In particular, when X is a continuous vector field and Γ is an invariant subset of X , our result implies that if Γ is non-spiralling then the Poincaré first return map associated Γ is also in ℝ an , exp .

Tenseness of Riemannian flows

Hiraku Nozawa, José Ignacio Royo Prieto (2014)

Annales de l’institut Fourier

We show that any transversally complete Riemannian foliation ℱ of dimension one on any possibly non-compact manifold M is tense; namely, M admits a Riemannian metric such that the mean curvature form of ℱ is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize some...

The absolute continuity of the invariant measure of random iterated function systems with overlaps

Balázs Bárány, Tomas Persson (2010)

Fundamenta Mathematicae

We consider iterated function systems on the interval with random perturbation. Let Y ε be uniformly distributed in [1-ε,1+ ε] and let f i ∈ C 1 + α be contractions with fixpoints a i . We consider the iterated function system Y ε f i + a i ( 1 - Y ε ) ⁿ i = 1 , where each of the maps is chosen with probability p i . It is shown that the invariant density is in L² and its L² norm does not grow faster than 1/√ε as ε vanishes. The proof relies on defining a piecewise hyperbolic dynamical system on the cube with an SRB-measure whose projection is the...

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