We discuss the multifractal analysis of a planar harmonic measure and the boundary rotation. We are especially interested in the sharp bounds for the local dimension of harmonic measure and the rate of rotation, the socalled universal spectrum. The theory is well studied for simply connected domains, with the help of the conformal mapping techniques. It is also well established for hyperbolic polynomial Julia sets, by the means of thermodynamical formalism.
We establish that the universal spectrum for the nonsimply connected domains are essentially the same as for the simply connected ones. Moreover, we show that the same sharp bounds are achieved on connected hyperbolic polynomial Julia sets. Thus a number of questions in geometric function theory, such as Brennan and Littlewood conjectures, are now can be considered as problems in polynomial dynamics.
This is joint work with P. Jones (Yale), N. Makarov (Caltech) and S. Smirnov (Geneva).
While the computer is a discrete device, it is often used to solve problems of a continuous nature. The field of Real Computation addresses the issues of computability in the continuous setting. As in the discrete case, we would like to define the notion of a computable subset of R^{n}. The definition we use has a computer graphics interpretation (in the case n=2), as well as a deeper mathematical meaning.
We show that the hyperbolic Julia sets are polytime computable in this model, which reinforces the intuitive understanding that hyperbolic Julia sets are `easy' to draw. Our computability results come in contrast to the Julia sets noncomputability results presented by Blum/Cucker/Shub/Smale. This discrepancy follows from the fact that we are using a different computability model.
If a is an irrational number, the Bruno function F is
defined by

The quadratic polynomial P_{a} : z® e^{2ipa}z+z^{2} has an indifferent fixed point at the origin. If P_{a} is linearizable, we let r(a) be the conformal radius of the Siegel disk and we set r(a)=0 otherwise.
We prove that the function a® F(a) + logr(a), which is welldefined on the set of Bruno numbers, has a continuous extension to R.
Soient G un groupe de Lie réel simple et connexe de rang réel r et G un réseau de G. Soit M une variété complexe compacte kaehlerienne de dimension complexe d. Nous montrerons que tout morphisme de G dans le groupe des difféomorphismes holomorphes de M a une image finie dés que r est strictement supÃ©rieur à d.
Let G be a simple connected real Lie group and G a lattice in G. Let M be a compact complex kaehler manifold. We will prove the following theorem: If the real rank of G is larger than the complex dimension of M, then any morphism from G into the group of holomorphic diffeomorphisms of M has a finite image.
(Travail commun avec X. Buff)
Nous expliquerons la construction, dans la famille des polynômes de degré 2 ayant un point fixe indifférent, de disques de Siegel D dont on contrôle la linéarisante f: D® D, au sens où l'on peut prescrire assez finement son degré de régularité au bord. Cela a pour conséquence, par exemple, qu'il existe des disques de Siegel dont le bord est une courbe plongée C^{n} mais pas C^{n+1}.
Let F be a foliation of class C^{2} of a compact manifold M, and g a Riemannian metric on TF. L. Garnett studied the diffusion semigroup along the leaves of F, acting on the continuous functions on M. She proved the existence of a probability measure on M, invariant by this semigroup (such a measure is called harmonic measure), and she developed the ergodic properties of harmonic measures.
In joint work with Victor Kleptsyn, we prove that on a minimal compact subset of a codimension 1 foliation of class C^{2} is supported a unique harmonic measure.
We construct the equilibrium measure for polynomiallike maps in Several Complex Variables. We prove that the measure is mixing, maximizes entropy and does not charge the critical set. We also study the distribution of preimages, and of repelling periodic points. This is a joint work with N. Sibony.
A rigidity theorem postulates that in a certain class of dynamical systems equivalence (combinatorial, topological, smooth, etc.) automatically has a higher regularity. I shall discuss several recent rigidity results for circle maps with singularities. The proofs use heavily the concept of renormalization, which will also be considered.
Using an approach introduced by Douady, namely approximations by polynomial vector fields, we study the dynamics of holomorphic maps which are small perturbations of a holomorphic map with a parabolic periodic point. We extend results of BodartZinsmeister and McMullen from symmetric perturbations to nonsymmetric perturbations.
In particular, we show that if {f_{l}} is an analytic family of rational maps over the unit disc such that f_{0} is geometrically finite and such that the critical orbit relations in the Julia set J(f_{0}) are preserved, then, for all radial perturbations of f_{0} except finitely many, J(f_{l}) tends to J(f_{0}) and the Hausdorff dimension of J(f_{l}) tends to that of J(f_{0}).
Joint with X. Buff.
This is a talk on joint work with Mikhail Lyubich.
A Henon map is an automorphism of C^{2} of the form f(x,y) = ( p(x)ay,x ). Friedland and Milnor showed that the only dynamically interesting automorphisms of C^{2} are Henon maps and their compositions.
We have been interested in deforming a Henon map f by deforming the underlying manifold. Hubbard and ObersteVorth introduced a pair of dynamically defined foliations F_{+} and F_{} for a given Henon map. Buzzard and Verma have proven some stability results using holomorphic motions along the leaves of F_{+}. One can not construct a meaningful deformation of a complex manifold by deforming only the leaves of a single foliation, but one can easily do so using the leaves of a pair of foliations. Hence one is led to ask whether a Henon map can be deformed using the pair of foliations F_{+} and F_{}.
We classify the critical locus of the foliations F_{+} and F_{} for a Henon map f(x,y) = ( p(x)ay,x ) when the Jacobian a of f is sufficiently small, assuming that p is a hyperbolic polynomial with connected Julia set and simple critical points. Using our classification of the critical locus for such Henon maps, we show that there is no conjugacy between two such Henon maps preserving both F_{+} and F_{}, unless the conjugacy is actually holomorphic or antiholomorphic on open sets, or the critical locus has a smooth boundary. Thus deformation using a pair of holomorphic foliations is too rigid.
In a joint work with M. Braverman we demonstrate the existence of quadratic polynomials whose Julia sets are noncomputable.