


SS17  Systèmes dynamiques complexes / SS17  Complex Dynamical Systems Org: X. Buff (Toulouse), A. Cheritat (Toulouse) et/and M. Yampolsky (Toronto)
 ARTUR AVILA, College de France

 ILIA BINDER, University of Toronto
Multifractal Analysis of Harmonic Measure

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).
 MARK BRAVERMAN, University of Toronto, Toronto, Ontario, Canada
On the Computability of Hyperbolic Julia Sets

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.
 XAVIER BUFF, Université Paul Sabatier, Toulouse
The Bruno function continuously approximates the size of
quadratic Siegel disks

If a is an irrational number, the Bruno function F is
defined by
F(a) = 
å
n ³ 0

a_{0} a_{1} ¼a_{n1}log 
1
a_{n}

, 

where a_{0} is the fractional part of a and a_{n+1}
is the fractional part of 1/a_{n}. The numbers a such
that F(a) < ¥ are called the Bruno numbers.
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.
 SERGE CANTAT, Université de Rennes 1, Bâtiments 2223 du campus de
Beaulieu, 35042 Rennes Cedex
Version kaehlerienne d'une conjecture de Zimmer/Kaehlerian
version of Zimmer conjecture

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.
 ARNAUD CHERITAT, Université Paul Sabatier, Laboratoire Emile Picard, 118 route
de Narbonne, 31062 Toulouse Cedex 4
Sur la régularité des disques de Siegel

(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}.
 BERTRAND DEROIN, University of Toronto
Unique ergodicity of codimension 1 foliations

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.
 TIEN CUONG DINH, Université ParisSud, Bât. 425  Mathématique, 91405
Orsay
Dynamics of polynomiallike maps in higher dimension

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.
 ADAM EPSTEIN, Warwick

 JOHN HUBBARD, Cornell and Marseille

 DIMA KHMELEV, University of Toronto
Renormalization and rigidity theory for circle maps with
singularities

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.
 TAN LEI, Université de Cergy Pontoise, 2 av. A. Chauvin, 95302
CergyPontoise
Dynamical convergence and polynomial vector fields

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.
 JOHN ROBERTSON, Dept. of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3, Canada
The Critical Locus of Henon Maps

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.
 MICHAEL YAMPOLSKY, University of Toronto
Noncomputable Julia sets

In a joint work with M. Braverman we demonstrate the existence of
quadratic polynomials whose Julia sets are noncomputable.

