Schedule
| Tuesday june 16th | |
| 9:30 - 10:30 | |
| 11:00 - 12:00 | Ricardo Pérez Marco
Minicourse: Germs of quadratic type, I |
| 12:15 | Lunch |
| 15:00 - 16:00 | |
| 16:30 - 17:30 | |
| Wednesday | |
| 9:30 - 10:30 | Robert Devaney
Minicourse: Cantor Bouquets in Transcendental Dynamics, II |
| 11:00 - 12:00 | |
| 12:15 | Lunch |
| 14:00 - 15:00 | |
| 15:00 - 16:00 | |
| 16:30 - 17:30 | |
| Thursday | |
| 9:30 - 10:30 | Ricardo Pérez Marco
Minicourse: Germs of quadratic type, II |
| 11:00 - 12:00 | Mitsuhiro Shishikura,
Minicourse: Local invariant sets of irrationally indifferent fixed points of high type, II |
| 12:15 | Lunch |
| 15:00 - 16:00 | Kingshook Biswas
Centralizers, Tube-log Riemann Surfaces and Hedgehogs |
| 16:30 - 17:30 | |
| Friday | |
| 9:30 - 10:30 | Ricardo Pérez Marco
Minicourse: Germs of quadratic type, III |
| 11:00 - 12:00 | Mitsuhiro Shishikura
Minicourse: Local invariant sets of irrationally indifferent fixed points of high type, III |
| 12:15 | Lunch |
| 15:00 - 16:00 | |
Abstracts
Lex Oversteegen On simple models of Julia sets
In this talk we will survey some old and new results regarding simple models for Julia sets.
We will adopt the convention that a space X is locally connect (LC) at a point p
if there exist arbitrary small open and connected sets containing p while a space is connected im kleinen at p
if there exists arbitrary small connected (but not necessarily open) neighborhoods at p.
Julia sets with nice local structure (i.e., LC) have been extensively studied and much is known. However, as is well known, non-locally connected Julia sets exist and their structure is still elusive. Extending a result by Kiwi, it has recently been shown that all connected Julia sets have an optimal locally connected model allowing for extensions of some results for locally connected Julia sets to the general case.
Unfortunately, in some key cases, the locally connected model is a single point. Nevertheless, such Julia sets may well be connected im kleinen at many points. In particular this is the case for certain Cremer Julia sets of positive area which were recently constructed by Buff and Chéritat. A main open problem is whether such Julia sets are arcwise connected.
Julia sets with nice local structure (i.e., LC) have been extensively studied and much is known. However, as is well known, non-locally connected Julia sets exist and their structure is still elusive. Extending a result by Kiwi, it has recently been shown that all connected Julia sets have an optimal locally connected model allowing for extensions of some results for locally connected Julia sets to the general case.
Unfortunately, in some key cases, the locally connected model is a single point. Nevertheless, such Julia sets may well be connected im kleinen at many points. In particular this is the case for certain Cremer Julia sets of positive area which were recently constructed by Buff and Chéritat. A main open problem is whether such Julia sets are arcwise connected.
Nuria Fagella Entire transcendental maps with two singular values and a persistent Siegel disk
We study the class of entire transcendental maps of finite order
with one critical point and one asymptotic value, which has exactly
one finite pre-image, and having a persistent Siegel disk. After
normalization this is a one parameter family
fa with a∈ℂ* which includes the semi-standard map
λ z ez at a=1, approaches the exponential map when a→0
and a quadratic polynomial when a→∞. We investigate the
stable components of the parameter plane (capture components and
semi-hyperbolic components) and also some topological properties of
the Siegel disk in terms of the parameter.
Robert Devaney Minicourse: Cantor Bouquets in Transcendental Dynamics, I
These talks will give a basic introduction to the structure
and properties of Cantor bouquets that arise in the dynamics
of entire functions. Particular emphasis will be on the
simplest case, the complex exponential family, though
other examples will be presented.
Files:
Slides/Bob.ppt,
Ricardo Pérez Marco Minicourse: Germs of quadratic type, I
We study a class of holomorphic germs with an indifferent irrational fixed point. Germs of quadratic type admit a semilocal sectorial renormalization. All their (convenient normalized) renormalizations are well approximated by a Moebius map. This allows to study through sectorial renormalization not only the linearization properties, but also their hedgehogs for non-linearizable germs. In that case we have a model for the topological and the dynamics on the hedgehog that can be analyzed in full.
Mitsuhiro Shishikura Minicourse: Local invariant sets of irrationally indifferent fixed points of high type, I
We discuss the structure of local and semi-local invariant sets for the class of functions
irrationally indifferent fixed points introduced by a joint work with H. Inou
(in particular, the results apply to quadratic polynomials).
It was shown that if we restrict to the rotation numbers of high type (i.e., those with
large continued fraction coefficients),
there exists a class of functions which is invariant under near-parabolic renormalization R.
As an application, we define the ``maximal hedgehog'' Λf which contains the critical orbit as well as all hedgehogs in Pérez Marco's sense. We will also show that Λf consists of arcs (hairs) which are disjoint except at the fixed point, and the dynamics acts as a rotation-like permutation among them; when the rotation number α satisfies Brjuno condition, then the boundary of Siegel disk is a Jordan curve, whose modulus of continuity can be characterized in terms of the continued fraction expansion of α; the critical point is on the boundary of the Siegel disk if and only if α belongs to the class H defined by Yoccoz; if the map is a quadratic polynomial, then the Julia set is locally connected at all repelling periodic points.
Talk 1. Parabolic and near-parabolic renormalization and the invariant class F1. We go through the definition of return maps and near-parabolic renormalization and Inou-S. result on the invariant class F1. The Truncated Checkerboard Pattern Ωf will be introduced. An important part of dynamics of f will be reinterpreted via Ωf.
Talk 2. Reconstructing (part of) f from Rf and the rotation combinatorics. As a first step, we will see how a part of the original dynamics f can be reconstructed from the renormalization Rf by gluing copies of Truncated Checkerboard Pattern ΩRf. In order to describe further steps, we study the combinatorics of rotations and their symbolic representation give rise to the index set An for the copies of Truncated Checkerboard Pattern ΩRnf. The definition of Λf will be given and its basic properties will be discussed.
Talk 3. Applications---hedgehogs, hairs and Siegel disks. Based on the reconstruction (of f from Rnf) in the previous talk, we will prove the results that are mentioned above.
As an application, we define the ``maximal hedgehog'' Λf which contains the critical orbit as well as all hedgehogs in Pérez Marco's sense. We will also show that Λf consists of arcs (hairs) which are disjoint except at the fixed point, and the dynamics acts as a rotation-like permutation among them; when the rotation number α satisfies Brjuno condition, then the boundary of Siegel disk is a Jordan curve, whose modulus of continuity can be characterized in terms of the continued fraction expansion of α; the critical point is on the boundary of the Siegel disk if and only if α belongs to the class H defined by Yoccoz; if the map is a quadratic polynomial, then the Julia set is locally connected at all repelling periodic points.
Talk 1. Parabolic and near-parabolic renormalization and the invariant class F1. We go through the definition of return maps and near-parabolic renormalization and Inou-S. result on the invariant class F1. The Truncated Checkerboard Pattern Ωf will be introduced. An important part of dynamics of f will be reinterpreted via Ωf.
Talk 2. Reconstructing (part of) f from Rf and the rotation combinatorics. As a first step, we will see how a part of the original dynamics f can be reconstructed from the renormalization Rf by gluing copies of Truncated Checkerboard Pattern ΩRf. In order to describe further steps, we study the combinatorics of rotations and their symbolic representation give rise to the index set An for the copies of Truncated Checkerboard Pattern ΩRnf. The definition of Λf will be given and its basic properties will be discussed.
Talk 3. Applications---hedgehogs, hairs and Siegel disks. Based on the reconstruction (of f from Rnf) in the previous talk, we will prove the results that are mentioned above.
Arnaud Chéritat Relatively compact Siegel disks with non-locally connected boundaries
I will adapt a construction of Pérez Marco to prove the existence of an injective holomorphic map defined on a simply connected subset U of C, having a Siegel disk compactly contained in U whose boundary is a pseudocircle.
Kingshook Biswas Centralizers, Tube-log Riemann Surfaces and Hedgehogs
We study the centralizers of irrationally indifferent nonlinearizable germs in the group Diff(C,0) and their relation to the invariant continua for the dynamics, which are called hedgehogs.
We first describe a general construction of dynamics due to Perez-Marco using translation flows on flat Riemann surfaces called "tube-log Riemann surfaces", which allows one to construct uncountable centralizers while simultaneously controlling the geometry of the hedgehogs obtained. We sketch how to use these techniques to construct hedgehogs of Hausdorff dimension one and hedgehogs containing smooth combs (homeomorphic images of a the product of a Cantor set and an interval).
Perez-Marco proved that commuting germs preserve the same unique family of hedgehogs. We prove a converse result: two germs preserving a common hedgehog must commute. As a consequence we obtain a bijective correspondence between families of hedgehogs and centralizers. We also show that the conjugacy class of a nonlinearizable germ is determined by its family of hedgehogs and its rotation number: any germ mapping a hedgehog of one germ to a hedgehog of another germ with the same rotation number must conjugate the two germs. These results are consequences of the rigidity result which may be thought of as a Schwarz lemma for hedgehogs: if a germ tangent to the identity preserves a hedgehog then it must be the identity.
We first describe a general construction of dynamics due to Perez-Marco using translation flows on flat Riemann surfaces called "tube-log Riemann surfaces", which allows one to construct uncountable centralizers while simultaneously controlling the geometry of the hedgehogs obtained. We sketch how to use these techniques to construct hedgehogs of Hausdorff dimension one and hedgehogs containing smooth combs (homeomorphic images of a the product of a Cantor set and an interval).
Perez-Marco proved that commuting germs preserve the same unique family of hedgehogs. We prove a converse result: two germs preserving a common hedgehog must commute. As a consequence we obtain a bijective correspondence between families of hedgehogs and centralizers. We also show that the conjugacy class of a nonlinearizable germ is determined by its family of hedgehogs and its rotation number: any germ mapping a hedgehog of one germ to a hedgehog of another germ with the same rotation number must conjugate the two germs. These results are consequences of the rigidity result which may be thought of as a Schwarz lemma for hedgehogs: if a germ tangent to the identity preserves a hedgehog then it must be the identity.
Saeed Zakeri Siegel disks in a family of entire maps
Let f: ℂ → ℂ be an entire map of the form
f(z)=P(z)eQ(z), where P and Q are polynomials of arbitrary degrees (we
allow the case Q=0). Building upon a method pioneered by Shishikura, we
show that if f has a Siegel disk of bounded type rotation number centered
at the origin, then the boundary of this Siegel disk is a quasicircle
containing at least one critical point of f. This unifies and generalizes
several previously known results
Helena Mihaljevic-Brandt Semiconjugacies and pinched Cantor bouquets
It is well known that the Julia set of a transcendental entire
function can be the whole complex plane.
It seems that yet, there is no example of such a map for which the topological
dynamics has been completely understood.
In this talk we will provide such a description for the map π sinh z.
More generally, we will present a result which enables us to describe the
Julia set of every function in a large class of maps as a quotient of the
Julia set of a (particularly simple) hyperbolic map in the same parameter
space.
We will show that for certain maps for which the Julia set is the whole plane (and many others), the Julia set can be described as a pinched Cantor bouquet, consisting of dynamic rays and their landing points.
We will show that for certain maps for which the Julia set is the whole plane (and many others), the Julia set can be described as a pinched Cantor bouquet, consisting of dynamic rays and their landing points.
Janina Kotus Cantor bouquets for non-entire mermorphic functions
For the basic families of entire functions like for e.g λ e^z or
λ sin z , λ∈ℂ*, the escaping set I(f) = {z∈ℂ: lim |f^n(z)| = ∞} is a Cantor bouquet.
Moreover, the escaping set can be written as I(f)=∩R>0
IR(f), where IR(f)={z: liminf |f^n(z)|
> R}. By contrast for the tangent family λ tan(z) the set
IR(f) is a union of Cantor sets. To find Cantor bouquets
invariant under some iteration of non-entire meromorphic function
f one has to consider the maps for which the asymptotic value is
eventually mapped onto infinity. We find a class of functions with
one or two asymptotic values eventually mapped onto infinity and
prove that the Julia set contains Cantor bouquets invariant under
some iterates of f. We give examples of functions with Cantor
bouquets of positive Lebesgue measure and estimate its Hausdorff
dimension. We also consider the family of Fatou functions
fλ(z)=z+e-z+λ, λ∈ℂ* and Re λ >1.
The map fλ has a Baker domain at ∞, the Julia set
J(fλ) is the Cantor bouquet and HD(I(fλ)) = 2. We also
consider the radial Julia set defined as Jr(fλ) := J(fλ)\I(f&lambda) and prove that 1< HD (Jr(fλ)) <2. Finally we show that the function λ → HD (Jr(fλ))
is real analytic for λ>1.
Xavier Buff Arithmetical hedgehogs
We will present a model for studying the topology of hedgehogs and test various conjectures. This model is based on previous work of Perez-Marco, Inou and Shishikura.