Warsaw, April 3 - 7, 2006
IN ONE-DIMENSIONAL DYNAMICS
Organizers: Feliks Przytycki and Juan Rivera-Letelier
Schedule (lectures are 50 minutes + ε for questions).
Monday, April 3
From 14:15, A course by M. Zinsmeister on SLE at Warsaw University
(PDF file of lectures)
Tuesday, April 4
Wednesday, April 5
Thursday, April 6
Friday, April 7
- Peter Haissinsky, Coarse conformal dynamics
Abstract. The aim is to introduce geometric methods
coming from the theory of negatively curved metric spaces to the
iteration of rational maps, and more generally to the dynamics
of finite branched coverings on metric spaces.
- Volker Mayer, Thermodynamical formalism for meromorphic functions of
Abstract. One of the main tools of the geometric and ergodic study of holomorphic
dynamically systems is the thermodynamical formalism. We present joint work
with Mariusz Urbanski in which we make this theory available for a very
general class of meromorphic functions and deduce several applications.
We first present this class of so called balanced meromorphic functions.
Then we pass to the thermodynamical formalism itself. The key point here is
that, given a balanced function, we can associate to it a Riemannian metric with
respect to which the transfer operator is well defined and can be controlled
thanks to Nevanlinna Theory.
- Mariusz Urbanski, Conformal Iterated Function Systems, generalizations
Abstract. The class of conformal graph directed Markov systems (with
countably many edges and finitely many vertices) will be defined. The
pressure function and conformal measures will be introduced and the
classification into regular, strongly regular, cofinitely regular and
irregular systems systems will be provided. An appropriate version of
Bowen's formula for the Hausdorff dimension of the limit set will be
discussed and the formula for the dimension of the closure of the
limit set will be provided. The θ parameter will be discussed.
Dynamical and geometrical rigidities will be mentioned. The space
of Iterated function systems (the set of
vertices is a singleton) acting on a phase space X will be
topologized and regularity properties of the Hausdorff dimension
function will be discussed. They include continuity and
real-analyticity. This theory will be applied to elliptic functions to
get lower bounds for the Hausdorff dimension of the Julia set. It will
be also applied for generalized polynomial-like maps to demonstrate
real-analyticity of pressure and real-analytic dependence of the
Hausdorff dimension on parameter if an appropriate family of such maps
is given. Applications to diophantine approximations and statistics of
return times will be mentioned. The case when the number of vertices is
also infinite will be discussed and applications to infinitely generated
Kleinian groups of Schottky type will be provided.
- Henk Bruin, Equilibrium states and Young tower constructions in
one-dimensional dynamics. (Joint work with Mike Todd)
Abstract: Young towers are being used for an increasing number of
applications; recently for the construction of equilibrium states for
specific potential functions.
Equilibrium states are invariant measures that maximize a certain
functional involving the potential function and entropy.
However, when using a Young tower construction, the resulting
equilibrium state may only be unique and maximal within the class of
measures that can be lifted to this specific Young tower. A priori, a
different Young tower may result in different equilibrium states.
In this talk I want to restrict attention to smooth maps on the
interval and discuss a canonical way of constructing Young towers using
the Hofbauer tower. I will show that, within the class of invariant
measures with positive Lyapunov exponent, equilibrium states are indeed
independent of the choice of the Young tower.
- Manfred Denker, A Local Limit Theorem for Beta-Transformations
Abstract. This is joint work with Aaronson, Sarig and Zweimueller. We establish
conditions for aperiodicity of cocycles (in the sense
of Guivarc'h and Hardy, obtaining, via a study of perturbations of transfer
operators, conditional local limit theorems. Our results
apply to a large class of Markov and non-Markov interval maps, including
- Guillame Havard, Non-finite equilibrium measures vs conformal measures in
iteration of a rational map on its Julia set
Abstract. When studying equilibrium measure for a given
potential φ, which means to find an invariant probability
measure which maximizes the pressure P(φ)$ of φ,
one may first look for a measure ω with Jacobian equal to
exp(P(φ)-φ). Then one may try to find an invariant
probability measure μ equivalent with ω. This will be
our equilibrium measure. It appears that sometimes this strategy
leads to a non-finite invariant measure. This could be the case if
for instance there are some parabolic points. In this talk I will
introduce a way to picture those invariant measures as equilibrium
measures. Then I will discuss unicity and existence of such
generalized equilibrium measures. The case of the iteration of a
parabolic rational map on its Julia set is the perfect field to
illustrate those notions.
- Mark Holland, Statistical Properties of Lorenz-like maps and
Abstract. Using methods of inducing we prove exponential decay of
correlations for a class of one-dimensional maps with singularities and
critical points. The motivation for studying such maps arises naturally
from the Lorenz equations.
Furthermore, we can use inducing to analyse the statistical properties
of the actual Lorenz attractor, where we prove that a Central Limit
Theorem and Almost-Sure-Invariance-Principal (approx by Brownian motion)
- Godofredo Iommi, Suspension flows over countable Markov shifts
Abstract. In this joint work with Luis Barreira, we introduce a notion of
topological pressure for suspension flows over countable Markov shifts. We
develop the associated thermodynamic formalism. In particular, we
establish a variational principle and find conditions that guarantee the
existence of equilibrium measures. As an application we present a
multifractal analysis for the entropy spectrum of Birkhoff averages.
- Janina Kotus, Invariant Measures for Meromorphic Misiurewicz
Abstract. We study the existence of finite absolutely continuous invariant
measures for meromorphic nonrecurrent maps whose Julia set is the whole
sphere. In the rational context, these hypotheses imply that such a
measure must exist. We show that it is not so for meromorphic maps
unless an additional condition on the behavior of the map, which can be
stated in terms of its Nevanlinna characteristic, is satisfied.
(joint work with G. Swiatek)
- Genadi Levin, Cantor attractors for critical circle covers
Abstract. I'll describe some tools in the proof
of the existence of such attractors (joint with G. Swiatek).
- Daniel Meyer, Thurston Maps, subdivisions, and conformal
Abstract. We consider postcritically finite branched coverings of the sphere, that are
expanding. There exists an invariant curve, so there is a natural Markov
partition for the dynamics. This eqips the sphere in a natural way with a metric
d. A theorem of Thurston gives a condition when such a map is equivalent to a
rational one. In this case the sphere with metric d is quasisymmetric to the
standard one. We prove a generalization of Thurstons theorem which gives a
precise bound for the dimension that can be attained under a qs map. This is
joint work with Mario Bonk.
- Juan Rivera-Letelier, Statistical properties of Topological Collet-Eckmann
Abstract. I will show that every Topological Collet-Eckmann
(non-uniformly hyperbolic) rational map on the Riemann sphere
possesses a unique conformal probability measure of minimal exponent
and that this measure is non-atomic ergodic and its Hausdorff dimension is
equal to the Hausdorff dimension of the Julia set. Moreover there is a
unique invariant probability measure absolutely continuous with respect
to this conformal measure; it has exponential decay of correlations and
satisfies Central Limit Theorem (on Hölder functions).
The existence of such a measure characterizes the Topological
(joint work with F. Przytycki)
- Michael Todd, Return time statistics in dynamical systems
Abstract. This is joint work with Henk Bruin. One way of finding the
return time statistics is to induce. A result of Bruin, Saussol,
Troubetzkoy and Vaienti shows that return time statistics to
balls/cylinders are often the same as those for a first return map.
This is useful since the statistics of the first return map is often,
though not always, well understood. Our method extends this idea to
inducing schemes. One result of this is that smooth interval maps (I,f)
such that almost every point has positive Lyapunov Exponent have
exponential return time statistics to balls with respect to an acip.
- Michel Zinsmeister, Rectifiability and Teichmüller theory