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# workshopapril03-abstracts

%

% ABSTRACTS FOR THE HOLOMORPHIC DYNAMICS WORKSHOP

%

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Avila, A: Hausdorff dimension of certain infinitely renormalizable Julia sets

(joint with Mikhail Lyubich)

We consider real quadratic polynomials which are infinitely renormalizable of

constant type. McMullen has asked if the corresponding Julia sets have always

Hausdorff dimension two. We show that this is not the case and give examples

for which the Julia set has Hausdorff dimension arbitrarily close to 1.

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Brakalova, M: On the Measurable Riemann Mapping Theorem

ABSTRACT: I'll discuss some recent developments and extensions of David's

results on the existence and uniqueness of solutions to the Beltrami Equation

with ||\mu||_{\infty}=1.

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Bullett, S: Dynamics of Hecke Groups, Chebyshev Polynomials and Matings

ABSTRACT: We show how Chebyshev polynomials arise in two quite different ways

in the study of matings between Hecke groups and polynomials (holomorphic

correspondences which are conjugate to Hecke groups on one part of the Riemann

sphere and conjugate to a polynomial on the complement). As a consequence we

obtain a description of the connectivity locus of the one parameter family of

scalar multiples of the nth Chebyshev polynomial, in the case that n is odd.

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Buff, X: Siegel disks of Quadratic Polynomials

ABSTRACT: We will present a technique introduced by Arnaud Cheritat

which provides powerful results regarding Siegel

disks of quadratic polynomials. In particular, one can prove

the existence of quadratic polynomials having a Siegel disk

with smooth boundary. One can also obtain good estimates for

the conformal radius of quadratic Siegel disks.

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Deroin, B: Levi-flat hypersurfaces in complex surfaces of positive curvature

ABSTRACT: Given a harmonic measure on a Riemann surface foliation of

a closed 3-dimensional manifold, we define its "normal class",

which measures how two leaves of the foliation are converging to

each other, using the brownian motion along the leaves. We bound

this normal class and we give applications to Levi-flat hypersurfaces

in complex surfaces of positive curvature.

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De Marco, L: Stability, Lyapunov exponents and metrics on the sphere

ABSTRACT: In any holomorphic family of rational maps, the Liapounoff exponents

as a function of parameters is shown to characterize stability. We give a

potential-theoretic formula for the Liapounoff exponent, and show how

the homogeneous capacity in C^2 is related to the study of conformal metrics

in the Riemann spehere.

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Dominguez, P: Dynamics of the sine family

(joint with Guillermo Sienra)

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Earle, C: Holomorphic contractibility of the normalized symmetric

homeomorphisms of the circle

ABSTRACT: Gardiner and Sullivan showed that the space of normalized

symmetric homeomorphisms of an oriented circle has a natural complex

structure making it a complex Banach manifold. We explain how to contract

it to a point in such a way that for each t in [0,1] the map f_t of

the space into itself is holomorphic. Our construction does not generalize

to the space of normalized quasisymmetric homeomorphisms.

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Epstein, A: Degenerate parabolic points and parameter space

discontinuities

ABSTRACT: A common strategy in holomorphic dynamics is to attempt to

decompose a complicated dynamical system into several simpler maps,

typically of lower degree. In certain cases, the inverse procedure may be

implemented by a surgical construction: for example, intertwining a pair

of quadratic polynomials to obtain a cubic polynomial, or mating them to

obtain a quadratic rational map. Any such construction yields a map

between subsets of parameter spaces. These maps are holomorphic

away from bifurcation loci. For one-parameter constructions, there are

known results assertion the global continuity of such maps. On

the other hand, we have learned to expect discontinuity in

multi-parameter settings. We present several mechanisms for producing

such discontinuities. These mechanisms are quite different, but they all

arise in connection with degenerate parabolic cycles. The existence of

suitably degenerate degenerate parabolics is in turn a property of

natural multi-parameter families.

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Geyer, L: Linearizability of irrationally indifferent fixed points

ABSTRACT: We present partial results supporting the conjecture that there are

no "exotic" Siegel discs, i.e. with non-Brjuno rotation numbers, for

polynomials and rational functions. Special attention will be given to the

case of cubic polynomials, where the critical part of parameter space is

amenable to (very limited) computer experiments.

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Gomez-Mont, X: On the geometry and dynamics of holomorphic flows

(joint with E. Ghys, J. Saludes; Ch. Bonatti, M. Viana and R. Vila)

ABSTRACT: We will explain how to apply Teichmuller Theory (quasiconformal

maps, Ahlfors-Bers Theory and the solution of the d-bar problem) to obtain

a dynamic division of a holomorphic foliation into 2 pieces: One with

automorphisms (Fatou component) and another one without symmetries (Julia

component). Then we will classify the Fatou components and give an

ergodicity result for the Julia set.

Restricting to a special class of polynomial differential equations

(Riccati equations) we will show that under a mild condition the

statistics of all leaves is the same. We use the Poincare metric on the

leaves to parametrize the leaf and we use the foliated geodesic and

horocyclic flows.

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Haissinsky: Tuning by surgery

ABSTRACT: In this talk, I will provide a constructive proof of the

tuning theorem which asserts the presence of small copies of the

Mandelbrot set in itself.

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Hruska, S: Hyperbolicity in the complex Henon family

ABSTRACT: The Henon map, H_{a,c}(x,y)=(x^2+c-ay,x) is a widely studied

family of maps with complicated dynamical behavior. Here we

regard H as a holomorphic diffeomorphism of C^2 and allow a,c to be

complex. Foundational work on the complex Henon family has been done by

Hubbard, Bedford and Smillie, Fornaess and Sibony, and others; however,

basic questions remain unanswered.

A good first step would be to understand hyperbolic Henon maps, which are

a class of maps which exhibit the simplest type of chaotic dynamics. A

complex Henon map is called hyperbolic if it is hyperbolic over its Julia

set, or equivalently, its chain recurrent set.

In this talk, we will describe the algorithm and results of a rigorous

computer program for testing whether for a given a,c, the complex

Henon map H_{a,c} is hyperbolic. Time permitting, we will also

discuss a (non-rigorous) program of Papadontanakis and Hubbard, which

draws pictures illuminating the rich and subtle dynamics of the complex

Henon family.

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Kameyama, A: Coding and tiling of Julia sets for subhyperbolic rational maps

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Kotus, J: Geometry and ergodic theory of nonrecurrent elliptic functions

ABSTRACT: We explore the class of elliptic functions whose critical points

in the Julia set are all nonrecurrent and whose omega-limit sets are

compact subsets of the complex plane. In particular, this class contains

hyperbolic, subhyperbolic and parabolic elliptic functions. Let h denote

the Hausdorff dimension of the Julia set of such a function. We construct

an atomless h-conformal measure m and we show that the h-dimensional

Hausdorff measure of the Julia set vanishes except when the Julia set is

the entire complex plane. The h-dimensional packing measure is always

positive and it is finite if and only if there are no rationally

indifferent periodic points. Furthemore, we prove the existence (and

uniqueness, up to a multiplicative constant) of a sigma-finite

f-invariant measure mu equivalent to m. This measure is then proved to be

ergodic and conservative. We identify the set of those points whose open

neighorhoods have infinite mu-measure, and show that infinity is not in

that set.

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Lei, T: Cui's extension of Thurston's theorem

ABSTRACT: This is to characterize topologically geometrically

finite rational maps. It is a powerful tool to construct access to to

perturbe maps with parabolic points.

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Levin, G: Universality and dynamics of unimodal maps with infinite criticality

(joint with Greg Swiatek)

ABSTRACT: The universality in one-dimensional dynamics is described

by fixed points of renormalization operators. We study the limiting

behavior of these fixed-point maps as the order of the critical point increases to infinity. It

is shown that a limiting dynamics exists, with a critical point that is

flat, but still having a well-behaved analytic continuation to a

neighborhood of the real interval pinched at the critical point. We study

the dynamics of limiting maps and prove their rigidity. In particular, the

sequence of fixed points of renormalization converges, uniformly on the

real domain, to a mapping of the limiting type, as the criticality

tends to infinity along the reals. (This generalizes our result announced

on the December workshop.) We prove also a straightening theorem for the

limiting maps.

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Makienko, P: Poincare series and Fatou conjecture

ABSTRACT: Let R be a rational map. A critical point c is called summable

if the series sum_n 1/(R^n)'(R(c)) is absolutely

convergent. Under certain topological conditions on the postcritical set

we prove that R can not be structurally stable if it has a summable

critical point c in J(R).

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Markovic, V: Isomorphisms of Teichmueller spaces and isometries of L^p

type spaces

ABSTRACT: We show that every biholomorphic map between Teichmuller spaces

(of Riemann surfaces which are of non exceptional type) must be a

geometric isomorphism. In particular, the group of automorphisms of the

Teichmuller space of a surface of non-exceptional type agrees with the

modular group of the surface.

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Okuyama, Y: The Siegel-Cremer problem from the Nevanlinna theoretical viewpoint

ABSTRACT: We study irrationally indifferent cycles of points or Jordan

curves for a rational function f - such a cycle is Siegel or Cremer, by

definition. We present a new argument from the viewpoint of Nevanlinna

theory. Using this argument, we give a clear interpretation of a

Diophantine quantity associated with an irrationally indifferent cycle.

This quantity turns out to be Nevanlinna-theoretical. As a consequence, we

show that an irrationally indifferent cycle is Cremer if this

Nevanlinna-theoretical quantitiy does not vanish.

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Oudkerk, R: The parabolic implosion and convergence to Lavaurs maps

ABSTRACT: We are interested in a convergent sequence of rational maps

f_n->f_0 where f_0 has a parabolic cycle. It can be

shown that either there is no "parabolic implosion" or else

we can pass to a subsequence such that we have f_n -> (f_0,g)

for some Lavaurs map g of f_0. This means that in some way

the "liminf" of the dynamical systems is a semi-group whose generators

include both f_0 and g, and that

"liminf J(f_n) \supseteq J(g) \supsetneq J(f_0)"

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Penrose, C: Regular and limit sets for holomorphic correspondences

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Perez, R: Geometry and combinatorics of Lyubich's principal nest

ABSTRACT: We present a description of admissible combinatorics for the

principal nest of a quadratic polynomial; this information helps for

instance, in the computation of exact moduli growth rates. As examples, we

characterize complex quadratic Fibonacci maps, construct complex

rotation-like maps and present a new dense autosimilarity result on the

Mandelbrot set.

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Rebelo, J: Dynamics of meromorphic vector fields and the geometry of

complex surfaces

ABSTRACT: Relations between the dynamics of certain meromorphic vector fields

and the geometry of complex surfaces.

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Rempe, L: Topology of Julia sets of Exponential Maps

ABSTRACT: We present a universal model for the dynamics of an exponential map

on its set of escaping points, which is a complete topological model

for the case of attracting or parabolic parameters. In fact, we show that

topologically the principle of renormalization is valid for these parameters.

We also remark on some results of rigidity of escaping dynamics and existence

of non-landing dynamic rays in the case where the Julia set is C.

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Rees, M: Views of the space of quadratic rational maps

ABSTRACT: The parameter space of quadratic rational maps is essentially

a space of two complex dimensions. There are many natural subspaces of

one complex dimension to consider. These subspaces tend to have

nontrivial topology, and even in a topological sennse, there is more than

one natural path from one rational map to another. When considering

rational maps as dynamical systems, there is, of course, much more

consider than just topological structure. I shall talk about how one

views one rational map in terms of another, probably with particular

reference to matings of polynomials.

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Rippon, P: On a question of Fatou

(joint with Gwyneth Stallard)

ABSTRACT: Let f be a transcendental entire function and let I(f) be the set

of points whose iterates tend to infinity. We show that I(f) has at

least one unbounded component. In the case that f has a Baker wandering

domain, we show that I(f) is a connected unbounded set.

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Shcherbakov, A: Generic properties of foliations determined by algebraic

differential equations on C^2.

A differential equation dw/dz=P_n(z,w)/Q_n(z,w) (where P_n and Q_n are

polynomials of degree at most n) can be extended to CP^2 as a holomorphic

foliation with singularities. The class of such equations is denoted by

A_n. A generic foliation from A_n has the line at infinity as a leaf. This

leaf has a non-trivial fundamental group. The corresponding holonomy

transformation group consists of germs of conforal mappings (C,0)->(C,0).

The orbit of a point z under the action of this holonomy group is the set

of images of z under the representatives of germs from this group, for all

germ having representatives defined at z.

If a group of germs of conformal mappings (C,0)->(C,0) is nonsolvable

then:

1) Its orbits are dense in sectors. That is, there is a finite set of

real analytic curves passing through 0 such that orbits are dense in

domains bounded by these curves.

2) The group is topologically rigid. That is, any homeomorphism which

conjugates it with another such group is holomorphic or antiholomorphic.

3) There is a countable set of germs whose representatives have isolated

fixed points away from 0.

Foliations in A_n have corresponding properties. More precisely, there

exists a real algebraic subset of the space A_n such that for any equation

from the complement:

1) Any leaf, other than the leaf at infinity, is dense in C^n.

2) The foliation is absolutely rigid. That is, if it is topologically

conjugate to some other foliation by a homeomorphism sufficiently close to

the identity then it is affinely equivalent to this foliation.

3) There exists a countable set of homologically independent complex limit

cycles.

There are other generic properties: a generic equation from A_n has no

cycle on the infinity leaf with identity holonomy map, and any leaf for a

generic equation is hyperbolic.

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Shen, W: Density of Axiom A in the space of real polynomials

ABSTRACT: In this joint work with Sebastian van Strien and Oleg Kozlovski,

we prove that for any d>=2, Axiom A maps are dense in the space of

real polynomials with real critical points of degree d, through a

rigidity approach.

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Singh, A: Transcendental entire functions whose Julia set is the complex

plane

ABSTRACT: We give a proceedure for constructing a new class of entire

function whose Julia set is the whole complex plane. We utilize the

properties of Schwarzian derivative and consider the critical and

asymptotic values of the function. These functions can be constructed to

be composite and non-periodic in nature.

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Stallard, G: Dimensions of Julia sets

ABSTRACT: We give examples of hyperbolic meromorphic and entire functions for

which the Julia sets have different Hausdorff and box dimensions.

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Urbanski, M: Fractal properties and ergodic theory of elliptic functions

ABSTRACT: Let q be the maximal order of all poles of an elliptic

function f. We will discuss the following results:

1. HD(J(f)) is greater than 2q/(q+1).

2. Hausdorff dimension of points escaping to infinity is at most

2q/(q+1).

We now consider the function f as mapping the torus T-f^{-1}(\infty)

onto T. Given a potential \phi, Holder continuous far from poles and of

the form u(z)+gamma\log|z-b|, gamma>2, u Holder continuous, near a

pole b, we define a pointwise pressure P(phi) and the corresponding

transfer operator L. Assuming that P(phi)> sup(phi) we will discuss the

following results:

1. P(phi)=log lambda$, where lambda is a positive eigenvalue of L.

2. There exists a unique real parameter c and a unique probability measure

m on J(f) such that (dm\circ f)/dm=exp(c-phi). This constant c

is equal to P(phi).

3. There is a unique f-invariant probability measure mu absolutely

continuous with respect to m. The dynamical system (f,mu) is metrically

exact, in particular mixing of all orders.

4. The transfer operator L acting on C(T) is almost periodic.

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Zdunik, A: Conformal measures and Hausdorff measures for the exponential

family

(joint with Mariusz Urbanski)

ABSTRACT: For a large class of maps a*exp(z) we study the dimension of

some natural, dynamically defined essential subset of a Julia set (It can

be understood as an analogue of a conical limit set). A conformal measure

supported on this special set is built and its ergodic properties are

studied. We also introduce a corresponding thermodynamical formalism.

Using this tool, we study the dependence of the above dimension on the

parameter a.

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