Nous étudions une classe L de fonctions de degré un du cercle, supposées de classe C^2 à l'exception de deux points où seule la continuité est exigée, et telles qu'elles soient constantes sur un des intervalles délimité par ces derniers. De plus sur des demi voisinages ouverts de ces points elles s'écrivent sous la forme x^l où l est un nombre réel positif appelé l'exposant critique de la fonction.
Nous montrons pour la sous-classe de L des fonctions dont le nombre de rotation est de type fini, l'existence d'une transition dans la géométrie du système lorsque l'exposant critique traverse 2. Cet résultat nous a permit de donner un exemple d'un flot de Cherry sur le tore avec ensemble quasi-minimal métriquement non-trivial.
Le cas plus général de fonctions en L avec nombre de rotation infini est ainsi considéré. Il devient pourtant plus délicat d'émettre des conjectures; on rencontre parfois des surprises dues à la présence de phénomènes paraboliques.
Nonarchimedean dynamics is a young branch of dynamics that runs parallel to (and has interactions with) the more classical field of complex dynamics. In this talk, I will discuss some of the difficulties that arise when doing analysis and dynamics over nonarchimedean fields, and describe ways of overcoming them. I will do this in the context of discussing recent progress in solving the equidistribution of preimages problem, which aims to provide a construction of a canonical invariant measure associated to nonarchimedean dynamical systems. In addition, I will highlight the connection between such dynamical equidistribution results and more recent arithmetic equidistribution results from Arakelov geometry.
This is a joint work with Cui Guizhen.
A parabolic point is a periodic point with multiplicity m that is at least 2. A perturbation will break the point into a certain number of points with total multiplicity m. A map f with parabolic points is often a common boundary parameter point of several distinct hyperbolic components. Some of them have a dynamics similar to that of f. We consider them as 'dynamically stable' perturbations of f. We will construct converging parameter rays within each stable perturbation. For this we will not use the usual approach of analysing the parametrization, instead we will use surgery to construct the path of maps with desired dynamical properties.
Questions about complex dynamical systems have traditionally been approached with techniques from analysis (complex or geometric). In the last 5 years or so, methods from arithmetic and algebraic geometry have played a central role -- and the result is an active new research area, the "arithmetic of dynamical systems" (to borrow the title of Silverman's textbook on the subject). The questions themselves have evolved, inspired by results from arithmetic geometry. In this talk, I will present joint work with Matt Baker, where we study "special points" within the moduli space of complex polynomial dynamical systems.
Jakobson's theorem says that a certain family of unimodal maps of the interval has a positive measure set of ``stochastic'' parameters for which there exist invariant measure absolutely continuous with respect to the Lebesgue measure. Luzzatto-Takahasi gave an effective estimate on the measure, but it was like $10^{-5000}$ for the quadratic family. We will present an alternative approach to Jakobson's theorem using complex extension and Yoccoz puzzle/parapuzzle techniques, and try to improve the estimates on the measure of stochastic parameters.
Given a branched topological covering $f:(S^2,P) \to (S^2, P)$ of the sphere by itself, with branch values contained in $P$, can $f$ be realized as a rational map? We give a positive criterion, a counterpart to the obstruction W. Thurston found in 1982. We show that, in the case that every periodic cycle in $P$ contains a branch point, $f$ is rational iff there is a metric spine $G$ for $S^2\setminus P$ so that $f^{-n}(G)$ conformally embeds inside $G$ for sufficiently large $n$.
Here, a map $p: G \to H$ between metric graphs is a $\textit{conformal embedding}$ if, for almost all $y \in H$,
\[ \sum_{f(x)=y} |f'(x)| \le k < 1. \] (The intuition is that $G$ conformally embeds inside $H$ if a slight thickening of $G$ conformally embeds as a Riemann surface inside a slight thickening of $H$.)
Since we construct a combinatorial object that exists for rational maps (rather than an obstruction that exists only for maps that are not rational), this provides a new object to study for rational maps. In particular, we can extract an $\textit{asymptotic stretch factor}$ by looking at the best constant $k$ in the conformal embedding, and speculate that it is related to the core entropy of $f$.
This is joint work with Kevin Pilgrim.
Il s'agit d'un travail en collaboration avec Charles Favre. Nous montrer que, dans l'espace des modules de polynômes de degré d, les paramètres Misirewicz à combinatoire fixée, ainsi que les paramètres hyperboliques possédant (d-1) cycles attractifs à multiplicateurs donnés s'équidistribuent vers la mesure de bifurcation. Notre démonstration repose sur le Théorème de Yuan d'équidistribution des points de petite hauteur et utilise de façon cruciale les résultats de transversalité d'Adam Epstein.
If F is an increasing homeomorphism of the real line with the property that F(x+1)=F(x)+1 for any x, then F induces an orientation preserving circle homeomorphism f. The average amount by which each point on the real line is translated under the action of F is called the translation number of F, similarly the average angle by which every point on the circle is rotated by the action of f is called its rotation number. In this talk we study the parameter space of certain two parameter family of analytic circle diffeomorphism with the help of rotation numbers.
A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set M) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system.
We shall discuss a new instance of this principle in terms of entropy. Indeed, recently W. Thurston defined the core entropy of the map f_c = z^2 + c as the entropy of the restriction of f_c to its Hubbard tree.
The core entropy changes very interestingly as the parameter c changes, and we shall relate such variation to the geometry of M. Namely, we shall compare the Hausdorff dimension of certain sets of external rays landing on veins of M to the core entropy of quadratic polynomials f_c along the vein.