17.45-18.15 Skorulski:
Metric properties of some family of transcendental meromorphic functions
Abstract:
We deal with the maps
$$f(z)=\frac{a\exp(z^p)+b\exp(z^{-p})}{c\exp(z^p)+d\exp(z^{-p})}$$
that have an asymptotic value eventually mapped onto infinity. We show
some metric properties of the Julia set e.g. estimate the Hausdorff
dimension of recurrent points and non recurrent as well, provide a
sufficient condition for the Julia set to have positive Lebesgue
measure and a condition for the measure to be zero. We also construct
a conformal measure.
Tuesday 22 July
9:30-10:20 Urbanski: 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:
\begin{enumerate}
\item $HD(J(f))>{2q\over q+1}$.
\item Hausdorff dimension of points escaping to infinity is $\le {2q\over q+1}$.\end{enumerate}
We now consider the function $f$ as mapping the torus
$Tf^{-1}(\infty)$ onto $T$. Given a potential $\phi$, H\"older continuous far from poles and of the form $u(z)+\gamma\log|z-b|$, $\gamma>2$, $u$ - H\"older 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:
\begin{itemize}
\item $P(\phi)=\log\lambda$, where $\lambda$ is a positive eigenvalue of $L$.
\item There exists a unique real parameter $c$ and a unique probability measure $m$ on $J(f)$ such that ${dm\circ f\over dm}=\exp(c-\phi)$. This constant $c$ is equal to $P(\phi)$.
\item 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.
\item The transfer operator $L$ acting on $C(T)$ is almost periodic.
\end{itemize}
10:20-10.50 (coffee: common room)
10:50-11:40 Shen:
Density of Axiom A for real polynomials with real critical points
(Mini Course)
Abstract:
11:50-12:40 Smania:
Renormalization theory (Mini Course)
Abstract
14:00-14:50 Shen:
Density of Axiom A for real polynomials with real critical points
(Mini Course II)
Abstract:
14:50-15:30 (tea: common room)
15:30-16:00 Comerford:A survey of results in random iteration
of rational maps
Abstract: We discuss what is known in a new branch of
complex dynamics known as
random iteration. This is a variant of the standard situation in
rational iteration where instead of considering the iterates of a fixed
rational function or polynomial, one allows the functions one considers
to vary at each step of the iterative process. Using the existing
theory as a guideline, one then seeks to examine which of the classical
results from complex dynamics carry over as generalzations to the new
setting and find counterexamples for those which do not. What one
generally finds is that with no additional restrictions on the sequences
of functions one considers, the deeper classical results such as
Sullivan's non-wandering theorem tend not to carry over. However, with
additionl restrictions on the dynamics such as hyerbolicity or semi-
hyperbolicity, many of the classical results can be generalized although
new techniques are required to prove them.
16:15-16:45 Vargas: Decay of Geometry and Invariant Measures
for Cubic Polynomials
Abstract:
We introduce the class of Fibonacci cubic real polynomials
whose critical points are strongly recurrent. These polynomials exhibit
decay of geometry as we have proved in a previous work. Here we use the
decay of geometry to prove the existence of an absolutely continuous
invariant measure.
17:00-17:30 Rempe: On a Question of Herman, Baker and Rippon
Concerning Siegel Disks
Abstract:
Herman, Baker and Rippon posed the question whether any
unbounded Siegel disk of an exponential map
$z\mapsto \lambda(\exp(z)-1)$ must contain the singular value
$-\lambda$ on its boundary. We give a positive answer
to this question.
17.45-18.15 Pinto: TBA
21.00-???? Problem session (room 26 of common room;
chaired by Kolyada and Misiurewicz)
Wednesday 23 July
9:30-10:20 Liverani:
Strong statistical properties of Anosov maps and flows - a functional
analytic approach (Mini Course I)
Abstract:
10:20-10.50 (coffee: common room)
10:50-11:40 Buzzi:Piecewise affine surface homeomorphisms with positive entropy
Abstract:
We analyze piecewise affine surface homeomorphisms with positive
topological entropy w.r.t. entropy, measures of maximum entropy and
periodic points. The key ingredient is a semi-uniform estimate on
local invariant manifolds, ie, an estimate which holds for a set of
positive measure w.r.t. any measure with large entropy. From this, we
are able to build a Markov structure and deduce our results.
This result extends previous works on the theme "hyperbolicity from
complexity" for non-uniformly expanding maps and (we hope) paves the
way for the analysis of general surface diffeomorphisms.
11:50-12:40 Avila: Reducible and non-uniformly hyperbolic quasiperiodic Schrodinger cocycles (joint with Raphael Krikorian)
Abstract:
The study of the spectral properties of the Schrodinger equation with quasiperiodic potential in dimension one is intimately related to a dynamical problem: the understanding of a a certain family of cocycles (parametrized by the energy) with values in SL(2,R) over a rotation of the circle. Using the wide range of techniques of the field, including complexification, renormalization, and local theory (KAM), we prove the following theorem (which can be seen as the analogue of Lyubich's ``regular or stochastic'' dychotomy for the quadratic family): if the potential is sufficiently smooth and if the frequency of the rotation satisfies an arithmetic condition of full measure then for almost every value of the energy the cocycle is either reducible (that is, conjugate to a constant) or non-uniformly hyperbolic. This result is connected to a rigidity statement whose proof involves getting ``a priori bounds'' for renormalization under convenient hypothesis. Among the spectral consequences of this result is a proof of a conjecture of Aubry and Andre on the measure of the spectrum of the Almost Mathieu operator.
14:00-14:50 Hasselblatt: Computing dimensions of hyperbolic sets from
those of stable and unstable slices (joint with Schmeling)
Abstract: As a topological counterpart to the Eckmann--Ruelle
conjecture we
conjecture that the fractal dimension of a hyperbolic Cantor set in a
dynamical system is the sum of the fractal dimensions of "typical" stable
and unstable slices. This talk presents significant progress towards
establishing this conjecture.
14:50-15:30 (tea: common room)
15:30-16:00 Vasquez: Gibbs cu-states and SRB measures
Abstract:
Alves, Bonatti and Viana proved the existence of SRB measures
for diffeomorphisms partially hyperbolic with mostly expanding
center-unstable direction. The main tool used there is the existence of
Gibbs cu-states.
In this work we prove many properties of such states and with respect to its
relationship with the SRB measures. More precisely
we describe their support and characterize the associated densities.
Also we show that the
methods used by Alves Bonatti and Viana give all Gibbs cu-states.
Moreover, we prove that
the Gibbs cu-states varies continuously with the diffeomorphism.
As consequence we conclude the statistical stability in many open classes of
diffeomorphisms. Also, we prove that every SRB measure is a Gibbs
cu-state.
16:15-16:45 Oliveira:
Equilibrium states for nonuniformly dynamical systems
Abstract:
17.00-17.30 Sands:
Negative Schwarzian derivative everywhere
Abstract: Any $C^3$ unimodal map with a non-flat critical
point
and all periodic orbits strictly repelling can be conjugated via a
real-analytic coordinate change to a map with negative Schwarzian
derivative everywhere.
17.45-18.15 J\"ager:
Quasiperiodically forced interval maps with negative Schwarzian
derivative
Abstract:
In the study of quasiperiodically forced systems
invariant graphs have a special significance: If the fibre maps are
all monotone inverval maps they occur generically as boundary
lines of compact invariant sets, every invariant ergodic measure
is associated to some invariant graph and the dynamical behaviour of
typical orbits is determined by the Lyapunov exponents of the adjacent
invariant graphs.
The presented work deals with systems of quasiperiodically forced
interval maps with negative Schwarzian derivative. The main result is
a classification with respect to the number of invariant graphs and
their stability (Lyapunov exponents). It turns out, that the
possibilities for the invariant graphs are exactly analogous to those
for the fixed points of the unperturbed fibre maps.
Thursday 24 July
9:30-10:20 Smania: Renormalization Theorey (Mini Course II)
Abstract:
10:20-10.50 (coffee: common room)
10:50-11:40 Vaienti: Invariant measures for multidimensional
non-uniformly expanding maps
Abstract:
11:50-12:40 Schmeling: Lyapunov exponents applied
Abstract:
We present some specific application of Lyapunov exponents in harmonic analysis, number theory and probability.
14:00-14:50 Kolyada:
Minimal maps and minimal spaces
Abstract:
Minimal maps are one of the main objects in topological dynamics.
One basic problem is whether a given space admits a minimal map or a
minimal homeomorphism. In this talk we consider this problem as well
as some topological properties of minimals maps.
14:50-15:30 (tea: common room)
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