Schedule for: 16w5129 - Analysis and Dynamics

A welcome drink will be served at the hotel.

Every planar dendrite gives rise to a lamination of the unit disc (via conformal mapping), and this lamination has a well-understood combinatorial description in the dynamical setting. I will discuss an analytic description of the laminations associated with trees whose complement is John (semi-hyperbolic dynamics). I will also discuss applications to combinatorial trees (Shabat polynomials and dessins d'enfants) and will speculate about random trees (CRT and the Brownian map).

(Joint work with B.Duplantier, H.Ho and B.Le).\\ If $f$ is a holomorphic and injective map from the unit disk into the complex plane normalized by $f(0)=0,\,f'(0)=1$ one defines its "integral means spectrum" as $p\mapsto \beta_f(p)$ where $\beta_f(p)$ is such that $$\int_0^{2\pi}\vert f'(re^{it})\vert^p dt\sim (1-r)^{-\beta_f(p)}$$ as $r\to 1$. The universal integral means spectrum is the supremum of the set of $\{\beta_f(p)\}$, $f$ over the set of all normalized conformal maps. It is well-known that if we restrict to bounded conformal maps one obtains a different spectrum. The first aim of this talk is to define a generalized integral means spectrum as $(p,q)\mapsto \beta_f(p,q)$ where $$\int_0^{2\pi}\frac{\vert f'(re^{it})\vert^p}{\vert f(re^{it})\vert^q} dt\sim (1-r)^{-\beta_f(p,q)}$$ allowing to unify the bounded/unbounded approach.\\ We put to use this new concept to study the generalized integral means spectrum of the whole-plane SLE; the case $q=2p$ corresponds to the external (bounded) case studied by Beliaev and Smirnov. There is a gap the latter paper: the proof fails for $p (Conference Room San Felipe)

We present an optimal estimate on the Fatou coordinates that plays a key role in the study of the dynamics of holomorphic maps near irrationally indifferent fixed points. The estimates have been established using quasi-confrormal techniques, and are a major improvement over known estimates obtained using classical methods of complex analysis. Time permitting, we discuss some of the applications of the new estimates.

In the moduli space of polynomial or rational maps of degree d, there exists a bifurcation measure which measures the unstability of the dynamics. I will discuss the equidistribution of the centers of the hyperbolic components towards that measure in various settings. These are joint works with T. Gauthier and Y. Okuyama.

Semihyperbolic rational maps of the Riemann sphere form a natural class of conformal dynamical systems. The aim of the talk is to provide a topological characterization of this class based on a single numerical invariant: the Ahlfors-regular conformal dimension.

Complex rational maps induce rich and interesting dynamics on the Riemann sphere. We consider what happens to the dynamics when a rational map tends to the boundary of moduli space, i.e. tends to a lower degree map. A typical example is given by a stretching quasiconformal deformation of Fatou sets. In this case, the limit can be described a piecewise linear map on a tree. We discuss an inverse problem and related questions on quasiconformal deformation of annuli.

In this talk, I will present an estimate for the dimension of a k-quasicircle which is the image of the unit circle under a k-quasiconformal mapping whose dilatation is supported on a union of horoballs located at least a hyperbolic distance R apart. The estimate is sharp up to a multiplicative constant. To motivate the proof, I will first discuss an analogous estimate for the growth of solutions of certain parabolic PDEs given by the Feynman-Kac formula.

Consider a conformal map from the unit disk onto a quasidisk. We determine a range of critical complex powers with respect to which the derivative is integrable. The results fit into the picture predicted by a circular analogue of Brennan's conjecture.

I shall discuss some recent developments on two closely related questions: How do projections affect Hausdorff dimensions of Borel sets in Euclidean spaces and what can one say about the Hausdorff dimension of the intersection of two sets, one fixed and the moved by translations and rotations? The methods rely on the Fourier transform.

I will describe recent results concerning the existence of some elements of thermodynamic formalism, such as topological pressure, Bowen's formula and conformal measures, for a large class of transcendental entire and meromorphic maps on the complex plane. This is a joint work with Bogusława Karpińska and Anna Zdunik.

Two infinitely renormalizable unimodal maps of bounded combinatorics which belong to the same topological conjugacy class have Cantor attractors which on small scales look the same. Colloquially, we say that topology determines geometry. This is what we have come to expect for one-dimensional maps. In this talk I will discuss recent surprising results that show that topology does not always determine geometry in the case of bounded combinatorics Lorenz maps.

We will discuss bifurcation loci within holomorphic families of endomorphisms of $P^k$ and show that their Hausdorff dimension is maximal near isolated Lattès maps. This is a joint work with Fabrizio Bianchi.

We study the dynamics of the family of meromorphic functions $f_{\lambda,\mu}(z)=\lambda e^z+\mu/z$ with non zero complex parameters. Each of these functions has one non omitted pole and infinitely many critical values which accumulate at zero. For certain parameters the functions have a cycle of period two of Baker domains, some of them have connectivity $\infty$ and others are simply connected.

We consider a class of critically tame elliptic function $f : \mathbb C \to \hat{\mathbb C}$. We give a construction of finite invariant measure $\mu$ absolutely continuous with respect to h-conformal measure for these maps, where where h is the Hausdorff dimension of the Julia set f. We establish the exponential decay of correlations, the Central Limit Theorem, and the Law of Iterated Logarithm with respect to the measure $\mu$. We also prove that $h_\mu(f) < \infty$.

We study properties of dynamical quantities while crossing the Mandelbrot set ${\cal M}$ by smooth curves passing by generic points with respect to the harmonic measure $\omega$. We prove that typical $c\in \partial {\cal M}$ with respect to $\omega$ can be approximated by Collet-Eckmann parameters with density and by hyperbolic parameters in a uniform way along every smooth curve traversing $\cal M$ at $c$. A geometric counterpart of this dynamical property is that every smooth curve passing through $c$ must lie outside of $\cal M$ at infinitly many scales of positive density. It is known that the Hausdorff dimension of Julia sets ${\cal J}_c$ on the boundary of $\cal M$, $c\in \partial {\cal M}\mapsto \HD({\cal J}_c)$ is discontinuous at every generic $c$ with respect to $\omega$. Yet, for almost every parameter $c\in \partial {\cal M}$ with respect to $\omega$ and every smooth curve $\gamma:[-1,1]\mapsto {\mathbb C}$ with the property that $c=\gamma(0)\in \partial {\cal M}$, there exists a set ${\cal A}_\gamma\subset [-1,1]$ having $0$ as a Lebesgue density point and such that that $\lim_{x\rightarrow 0} \HD({\cal J}_{\gamma(x)})=\HD({\cal J}_{c_0})$.}