Resultados de F. Cano sobre una conjetura de Thom.
Results and open questions on some invariants measuring the dynamical complexity of a map
Let f be a continuous map on a compact connected Riemannian manifold M. There are several ways to measure the dynamical complexity of f and we discuss some of them. This survey contains some results and open questions about relationships between the topological entropy of f, the volume growth of f, the rate of growth of periodic points of f, some invariants related to exterior powers of the derivative of f, and several invariants measuring the topological complexity of f: the degree (for the case...
Resurgence in a Hamilton-Jacobi equation
We study the resurgent structure associated with a Hamilton-Jacobi equation. This equation is obtained as the inner equation when studying the separatrix splitting problem for a perturbed pendulum via complex matching. We derive the Bridge equation, which encompasses infinitely many resurgent relations satisfied by the formal solution and the other components of the formal integral.
Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé
Henri Poincaré avait déjà remarqué que les variétés stable et instable du pendule perturbé, défini par l’hamiltonienne coïncident pas lorsque que le paramètre n’est pas nul, mais qu’on peut leur associer un même développement formel divergent en puissance de . Cette divergence est ici analysée au moyen de la récente théorie de la résurgence, et du calcul étranger qui permet de trouver un équivalent asymptotique de l’écart des deux variétés pour tendant vers zéro - du moins cela est-il montré...
Retracts, fixed point index and differential equations.
Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions.
Return time statistics for unimodal maps
We prove that a non-flat S-unimodal map satisfying a weak summability condition has exponential return time statistics on intervals around a.e. point. Moreover we prove that the convergence to the entropy in the Ornstein-Weiss formula enjoys normal fluctuations.
Reversibility in the diffeomorphism group of the real line.
Reversible Anosov diffeomorphisms and large deviations.
Reversible complex Hénon maps.
Riemann map and holomorphic dynamics.
Right closing almost conjugacy for G-shifts of finite type
A G-shift of finite type (G-SFT) is a shift of finite type which commutes with the continuous action of a finite group G. For irreducible G-SFTs we classify right closing almost conjugacy, answering a question of Bill Parry.
Rigid reparametrizations and cohomology for horocycle flows.
Rigidité de quelques groupes de germes de difféomorphismes.
Using the description of non solvable dynamics by Nakai, we give in this paper a new proof of the rigidity properties of some sub-groups of Diff(C, O). The Cinfinity case is also considered here.
Rigidité différentiable des groupes fuchsiens
Rigidité du flot géodésique de certaines nilvariétés de rang deux
Rigidity and inflexibility in conformal dynamics.
Rigidity and renormalization in one dimensional dynamical systems.
Rigidity of centralizers of diffeomorphisms
Rigidity of critical circle mappings I
We prove that two critical circle maps with the same rotation number in a special set are conjugate for some provided their successive renormalizations converge together at an exponential rate in the sense. The set has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of critical circle maps with the same rotation number that are not conjugate for any . The class of rotation numbers for which such examples exist contains...
Rigidity of harmonic measure
Let J be the Julia set of a conformal dynamics f. Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out to be equivalent to the existence of a conformal change of variable which reduces the dynamical system...