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workshopapril03-abstracts

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% 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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