## IHP

Institut Henri Poincaré

## Breaking parabolic points along stable directions

Type:
Type:
Site:
Date:
11/10/2013 - 10:30
Salle:
05
Orateur:
LEI Tan
Localisation:
Université d'Angers
Localisation:
France
Résumé:

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.

## Complex and arithmetic dynamics in dimension 1

Type:
Type:
Site:
Date:
13/09/2013 - 10:00
Salle:
05
Orateur:
Laura De Marco
Localisation:
Université Northwestern
Localisation:
États-Unis
Résumé:

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.

## A Proof of Jakobson’s Theorem via Yoccoz puzzles and the measure of stochastic parameters

Type:
Type:
Site:
Date:
13/09/2013 - 11:15
Salle:
05
Orateur:
Mitsuhiro Shishikura
Localisation:
Université de Kyoto
Localisation:
Japon
Résumé:

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.

## Detecting rational maps

Type:
Type:
Site:
Date:
05/07/2013 - 17:15
Salle:
05
Orateur:
Dylan Thurston
Localisation:
Université de l'Indiana
Localisation:
États-Unis
Résumé:

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.

## Distribution des polynômes postcritiquement finis

Type:
Type:
Site:
Date:
05/07/2013 - 15:00
Salle:
05
Orateur:
Thomas Gauthier
Localisation:
Université d'Amiens
Localisation:
France
Résumé:

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.

## On the boundaries of the Arnold tongues

Type:
Type:
Site:
Date:
28/06/2013 - 10:30
Salle:
05
Orateur:
Kuntal Banerjee
Localisation:
Université Paris-Est - Créteil
Localisation:
France
Résumé:

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.

## An entropic tour of the Mandelbrot set

Type:
Type:
Site:
Date:
21/06/2013 - 11:00
Salle:
201
Orateur:
Giulio Tiozzo
Localisation:
Université Harvard
Localisation:
États-Unis
Résumé:

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.

## Estimation de la covariance de matrices aléatoires

Type:
Site:
Date:
18/06/2013 - 10:30 - 12:00
Salle:
3B081
Localisation:
Université de Marne-la-vallée
Localisation:
France
Orateur:
YOUSSEF Pierre

## Plumbing construction and slices of Quasifuchsian space

Type:
Type:
Site:
Date:
31/05/2013 - 10:30
Salle:
05
Orateur:
Sara Maloni
Localisation:
Université Paris 11
Localisation:
France
Résumé:

Given a surface S, Kra's plumbing construction endows S with a projective structure for which the associated holonomy representation depends on some complex `plumbing parameters'. This construction gives elements of the Maskit slice, a well-known slice of the boundary of the Quasifuchsian space QF(S). In this talk, after reviewing basic results on Kleinian groups, Dehn--Thurston coordinates, complex projective structures and Maskit slice, we will describe the plumbing construction and a more general plumbing construction, called the c-plumbing construction, which builds elements in a particular slice of the Quasifuchsian space. Time permitting, we will describe some properties of these slices and we will show some pictures.

## Extensions naturelles et entropie des alpha-fractions continues

Type:
Type:
Site:
Date:
24/05/2013 - 10:30
Salle:
05
Orateur:
Wolfgang Steiner
Localisation:
Université Paris 7
Localisation:
France
Résumé:

Les $\alpha$-fractions continues de Nakada, où $\alpha$ est un nombre réel compris entre $0$ et $1$, constituent une généralisation des fractions continues régulières. La transformation de Gauss $x \mapsto 1/x - [1/x]$ sur l'intervalle $[0,1)$ est remplacée par la transformation $T_\alpha: x \mapsto 1/|x| - [1/|x|+1-\alpha]$ sur l'intervalle $[\alpha-1,\alpha)$. On peut associer à cette transformation une extension naturelle dont le domaine est une union (finie ou infinie) de rectangles, et l'entropie de $T_\alpha$ est proportionnelle à la réciproque de la mesure de ce domaine. Les rectangles peuvent être décrits par les $T_\alpha$-orbites des extremités de l'intervalle $[\alpha-1,\alpha)$. Pour $\alpha \geq \sqrt 2-1$, il s'agit d'une union d'au plus 3 rectangles, alors que le domaine est de nature fractale pour $\alpha < \sqrt 2-1$. Les $T_\alpha$-orbites de $\alpha-1$ et de $\alpha$ se rejoignent ultimement pour $\alpha$ dans une union dénombrable de sous-intervalles de $[0,1]$ dont le complément est de dimension de Hausdorff $1$. Bonanno, Carminati, Isola et Tiozzo ont établi une relation entre ce complément et l'intersection de la frontière de l'ensemble de Mandelbrot avec l'axe réelle.