
Primitivo Belén AcostaHumánez
( Fundación Universitaria Konrad Lorenz  FUKL )Homomorphism in some discreet dynamical systems
In this presentation we will show Homomorphism in three particular cases of discreet dynamical systems: the polynomials of Chebyshev, the sequence of Fibonacci and the geometric iteration. These homomorphism can be used to simplify calculations in the composition of these functions.

Diana Aldea Mendes
( ISCTE Lisbon )Kneading theory for twodimensional skewproduct maps
This is a joint work with J. Sousa Ramos. We consider skewproduct maps (or triangular maps) of the real plane and we construct a kneading theory and a Markov transition matrix for them by doing some tensor products between the corresponding onedimensional invariants associated with the basis and fiber map. An immediate consequence is the rigorous computation of the topological entropy and the determination of the Parry measure of these kind of twodimensional maps. During this work we only consider periodic and preperiodic points. Some examples are provided (Baker map, KaplanYorke map, etc.).

Bill Basener
( Rochester Institute of Technology )Transverse Disks, Prime Knots, and Nielson Theory
We present an overview of recent results involving transverse disks; cross
sections that are a disk instead of a closed manifold. We show that every
nonsingular differentiable flow has a global transverse disk, give a new
short proof that every manifold admitting a nonsingular flow has Euler
Characteristic zero, and show that Guittierrez's prime knotting property
holds for transitive flows and not just minimal ones. We present several
conjectures and discuss avenues for future research involving assymptotic
cycles and Nielson Theory.

Sergey Bezuglyi
( University of New South Wales (Sydney), Institute for Low Temperature Physics (Kharkov) )Topologies on groups of transformations in Borel and Cantor
dynamics.
This is a joint work with A.H.Dooley (Sydney) and J.Kwiatkowski (Torun). Motivated by the well known notions of the weak and uniform topologies in ergodic theory, we define and study topologies on the group Aut(X) of all Borel automorphisms of a standard Borel space X and on the
group Homeo(Y) of all homeomorphisms of a Cantor set Y.
We discuss the following topics in Borel and Cantor dynamics:
1) density of periodic transformations with respect to the defined topologies;
2) transformations of rank 1 and their closures;
3) Bratteli diagram for aperiodic Borel automorphisms;
4) minimal and mixing homeomorphism and their closures in the weak topology.

Andrzej Bis
( University of Lodz, Poland )Modelling minimal foliated spaces
with positive entropy
We formulate and discuss the following problem:
Problem. Given a compact metrizable space Z, find a compact
minimal foliated space (M,F) which is modelled on Z and has
positive entropy.
We show that the suspension construction of foliated space
cannot provide solution of problem for Z=S, the Sierpinski gasket in the plane. The situation becomes quite different
when one replace the Sierpinski gasket by Sierpinski carpet
or some its generalisations (called here Sierpinski sets).
We get some related results for Menger curve.

Alexander Blokh
( UAB )Attractors for graphcritical
rational maps (joint with M.
Misiurewicz)
We call a rational map graph critical if
each critical point either belongs to an
invariant finite graph, or has minimal limit set,
or is nonrecurrent and has the limit set
disjoint from the above introduced graph. We prove that
for any conformal measure either for almost every
point of the Julia set its limit set coincides with
the Julia set, or for almost every point of
the Julia set its limit set coincides
with the limit set of a critical point.

Phillip Boyland
( University of Florida )Topological obstructions for Dynamics of Fluid Flows
Many dynamical concepts have their origins in fluid flows. This raises a question: Are fluid flows a restricted class of dynamical systems, or can they display any possible dynamics (up to conjugacy? Any answers will, of course, be model and dimension dependent. We discuss this question mainly for 2D timeperiodic flows of Stokes and Euler type using a combination of topological methods from the ThurstonNielsen theory and analytical methods from the stability theory of periodic linear ODE's.

Zoltan Buczolich
( UNT and Eotvos University, Budapest )Convergent and divergent ergodic averages
In this talk I give a (mostly personal) history of
results related to convergence and divergence
of ergodic and ergodic type averages.
The talk concludes with mentioning
the most recent joint result
with Dan Mauldin, according to which there is
an irrational rotation $T_{\theta}$ of the unit circle $\T$
and a function $f\in L^{1}(\T)$ for which
$\frac{1}N\sum_{n=1}^{N}f(T_{\theta}^{n^{2}}x)$ diverges almost
everywhere. This answers a question of Bourgain.

Alexei Cheviakov
( Queen's University at Kingston )New exact nonsymmetric 3D MHD equilibria
In this talk I will present new methods of constructing exact
threedimensional solutions to the MHD equilibrium equations in isotropic
and anisotropic case. The new classes of solutions are not obliged to have
any geometrical symmetries, and can have various topologies  have
continuous families of spherical or toroidal magnetic surfaces or be dense
in 3D regions.
Examples of different types of analytical solutions will be presented, and
their properties and applications will be discussed.

Manav Das
( University of Louisville )IFS of finite type
IFS of overlapping similititudes have been studied for some
time. One special class, the IFS of finite type, was introduced by
SzeMan Ngai and Yang Wang. This was generalized to the graph directed
setting by Das and Ngai. In this talk, we discuss its relation to the
well known open set condition (OSC). This talk is based on joint work
with Gerald Edgar.

Rafael de la Llave
( Univ. of Texas at Austin )Geometric mechanisms for instability in Hamiltonian
systems
We describe a mechanism that produces orbits
which transverse a resonance in a near integrable
hamiltonian system.
We also describe the tools for a rigorous
verification of the existence of this
mechanism in concrete examples.
This is joing work with A. Delshams and
M. T. Seara.
A preprint is available
from www.ma.utexas.edu/mp_arc

Amadeu Delshams
( Universitat Politecnica de Catalunya (Barcelona) )Orbits of unbounded energy in generic quasiperiodic perturbations
of geodesic flows of certain manifolds
We show that certain mechanical systems, including geodesic flow
plus a quasiperiodic perturbation by a potential, have orbits of
unbounded energy.
The assumptions we make in the case of geodesic flows are: a) the
metric and the external perturbation are smooth enough, b) the
geodesic flow has orbits that satisfy some mild hyperbolicity
conditions, c) the frequency of the perturbation is Diophantine,
d) the external potential satisfies some nondegeneracy conditions
depending on the periodic orbits considered in b). The assumptions
on the metric are $C^2$ open and are known to be dense on many
manifolds. The assumptions of the potential fail only in infinite
codimension spaces of potentials.
The proof is based on geometric considerations of invariant
manifolds and their intersections, like the scattering map of
normally hyperbolic invariant manifolds, as well as standard
perturbation theories (averaging, KAM and Melnikov techniques).
This is a joint work with Rafael de la Llave and Tere M. Seara

Tomasz Downarowicz
( Wroclaw University of Technology )Entropy of a doubly stochastic operator  uniqueness of a good
definition.
We study doubly stochastic operators on $L^\infty(\mu)$.
We mention several attempts (existing in the literature) to define
entropy $h_\mu(T)$ of such operators with respect to the measure.
Having established an elementary property of such operators, which we
call ``asymptotic lattice stability'' we prove not only that all these
notions
are actually the same, but that there is a system of natural axioms, which
uniquely determines what entropy of a doubly stochastic operator must be.
We introduce another possible definition, which allows to interpret this
entropy as the rate of exponential growth of information.
The most striking observation is that there are examples of positive
entropy doubly stochastic operators which do not admit pointwise
generated factors, i.e., the entropy captures a purely ``operator'' type
dynamics.

Jorge Duarte
( ISEL Lisbon )Topological invariants in forced piecewiselinear FitzhughNagumolike systems
This is a joint work with J. Sousa Ramos.
Mathematical models for periodicallyforced excitable systems arise in many biological and physiological contexts. In previous papers, chaotic dynamics of a forced piecewiselinear FitzhughNagumolike system under large amplitude forcing was identified by H. G. Othmer and Min Xie. In this work we study the chaotic behavior of a special type of canonical return maps for a singular system, in some regions of parameter space. Using kneading theory we characterize the topological entropy and another topological invariant (to distinguish isentropic maps) of the singular system and we study the variation of this topological invariants with the parameters A, Theta and T.

Zhaosheng Feng
( Texas A&M University )On Traveling Wave Solutions and
Proper Solutions To the TwoDimensional
BurgersKortewegde Vries Equation
In this paper, the stability of a twodimensional
autonomous system is analyzed, which indicates that
under some particular conditions, the twodimensional
BurgersKortewegde Vries (2DBKdV) equation has a
unique bounded traveling wave solution. Then by using
the qualitative theory of differential equations,
the traveling solitary wave solution
to the 2DBKdV
equation is established. At the end of the paper,
the asymptotic
behavior of the proper solutions of
the 2DBKdV equation is presented.

Robbert Fokkink
( Delft University )Doubly asymptotic leaves in 1D hyperbolic attractors
Barge and Diamond classify 1D hyperbolic attractors by the number of cliques of asymptotic leaves, with interesting cliques containing at least three leaves. We show how to unzip leaves of the attractor, creating an arbitrary number of asymptotic pairs. This is joint work with Lex
Oversteegen and Jan Aarts

Lukas Geyer
( University of Michigan )Linearization at irrationally indifferent fixed points
Douady conjectured that irrationally indifferent fixed points of rational maps (of degree at least 2) are linearizable iff the rotation number satisfies the Brjuno condition. This conjecture is motivated by sharp results of Brjuno, Ruessmann and Yoccoz, which imply that it is true for quadratic polynomials. Though PerezMarco showed that the conjecture is "generically" true, it is still open even for cubic polynomials. We will present partial results for certain classes of rational mappings.

Ellina Grigorieva
( TWU )On Construction and Properties of Attainable Sets of a Dynamical Microeconomic Model
In this work, the presented nonlinear model of the process of production and sales of a perishable consumer good can be controlled either by the rate of production or by the price of the good. Attainable sets of corresponding controlled systems are investigated and the boundaries of these sets are studied. Since it has been proven that only bangbang controls with at most two switchings can lead trajectories to the boundary of the attainable sets, the boundaries of the sets can be constructed. Shapes of attainable sets for different parameters of the model will be demonstrated using MAPLE.
Keywords: Attainable set, nonlinear controlled system, microeconomic dynamical model

Roland Gunesch
( University of Leipzig )Counting periodic orbits of the geodesic flow
We establish an asymptotic formula for the number of homotopy classes of
periodic orbits for the geodesic flow on a manifold of nonpositive
curvature. This extends a celebrated result of G.A. Margulis to the
nonuniformly hyperbolic case and strengthens previous results by G.
Knieper. It is the most precise formula of this type that exists.
While proving this result, we also manage to carry out Margulis'
construction of the measure of maximal entropy without requiring strong
hyperbolicity.

J.T. Halbert
( University of Maryland, College Park )The Dynamics of a TaffyPulling Machine
Early investigators of diffeomorphisms naturally focused on the simplest case first: uniformly
hyperbolic maps. R.V. Plykin demonstrated in 1974 that nontrivial hyperbolic attracting sets exist
for some of these maps. Does the type of attractor he found (called a Plykin attractor) arise in
connection with a physical system? We hope to show that the answer is yes. We are currently studying
the dynamics of a taffypulling machine. We hope to show that the action of this machine on taffy
leads naturally to a diffeomorphism of an open set in the plane that has a Plykin attractor.

Jane Hawkins
( University of North Carolina at Chapel Hill )Parametrized Dynamics of Weierstrass Elliptic P Functions
We discuss properties of the Julia and Fatou sets of Weierstrass elliptic P functions, parametrized by the lattice and the invariants g2 and g3. We give examples of Lebesgue ergodic maps as well as conditions under which the Julia set is not the whole sphere.

Nicolai Haydn
( University of Southern California )The Distribution of the Measure of Cylindersets for NonGibbsian Measures
For ergodic measures the theorem of ShannonMcMillanBreimann asserts that
the metric entropy is given by the exponential decay rate of the
measures of cylinder neighbourhoods of points almost everywhere.
The distribution of the measure of these cylinder neighbourhoods
has extensively been studiedn in the case when the underlying measure
is Gibbsian involving some potential. We prove that for measures that
are not Gibbs one still obtains a Central Limit Theorem provided the
measure satisfies some mixing properties ($(\phi,f)$mixing). In fact
we show that the convergence is algebraic.
Using our approach we can also prove a Weak Invariance Principle
as well as a CLT for reentry times.

Charles Holton
( University of Texas at Austin )Unique desubstitution for substitution tilings
We consider aperiodic substitution tiling systems in d dimensions which
are closed under the action of the Euclidean group and minimal under the
subgroup of translations. The pinwheel tiling is one such example. The
main result is that for such a system the substitution is a homeomorphism.
This extends Mosse`s result for d=1 and Solomyak`s result for the
translationally finite case.

Yong Hou
( University of Iowa )Rigidity of Low Dimensional Group Action
rigidity of infinite volume 3manifolds with sectional curvature $b^2\le K\le 1$ and generalize the rigidity of Hamenstädt or more recently BessonCourtoisGallot, to 3manifolds with infinite volume and geometrically infinite fundamental group.

Roman Hric
( Matej Bel University, Slovakia and Instituto Superior Tecnico, Portugal )Characterization of minimal sets and nonhomogeneous minimal sets
Recent results on topological characterization of minimal sets in one dimension  on graphs and dendrites  will be presented. In connection with this, realization of nonhomogeneous minimal sets in low dimensions will be discussed as well.

Jun Hu
( The City Univ. of New York )Monotonicity of entropy and Ruelle operator
In this talk, we first explain the proof of Tsujji of the
monotonicity of entropy for the logistic family of real quadratic
polynomials. And then we do investigations on parallel results
for a special oneparameter family of real cubic polynomials
with negative Schwarzian derivatives.

Lois Kailhofer
( Alverno College )A classification of inverse limit spaces of tent maps with periodic critical points
We work within the oneparameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps $f_a$ and $f_b$ with periodic critical points, we show that the inverse limit spaces $(I,f_a)$ and $(I,f_b)$ are not homeomorphic when $a \neq b$. To obtain our result, we define topological substructures of a composant, called "wrapping points" and "gaps", and identify properties of these substructures preserved under homeomorphisms.

Robert Kallman
( University of North Texas )A Question of Ulam on the Full Permutation Group of the Natural Numbers
In 1960 S. Ulam asked if every Polish group can be algebraically embedded into the full permutation group of the natural numbers. This is answered in the negative by showing that there is no such algebraic embedding for the homeomorphism group of the real line.

Linda Keen
( Lehman College and Graduate Center, CUNY )Degenerate domains for Random Iterated Function Systems
\documentclass{article}[12pt]
\begin{document}
\title{Degenerate domains for Random Iterated Function Systems}
\begin{abstract}
In joint work with Nikola Lakic, we consider iterated function systems formed from the holomorphic family ${\mathcal H}ol(\Delta,X)$ where $\Delta$ is the unit disk and $X \subset \Delta$ is a subdomain. The subdomain $X$ is called {\em degenerate} if all limit functions are constant. For $X$ relatively compact in $\Delta$, Lorentzen and Gill showed that $X$ is degenerate and that every iterated system has a unique limit. For noncompact subdomains $X$, Beardon, Carne, Minda and Ng found sufficient conditions on the geometry of $X$ to guarantee that $X$ is degenerate. We prove that these conditions are also necessary. We also show that in the noncompact case an iterated system may have more than one limit.
\end{abstract}
\end{document}

James Keesling
( University of Florida )Adding Machines in the Quadratic Family of Functions
Authors: James Keesling and Louis Block
It is wellknown that adding machines occur in the quadratic family
of functions. The places where these are known to occur are where the quadratic
family is infinitely renormalizable.
It is more natural to study minimal sets that are slightly different than adding
machines in the quadratic family. We call these objects almost adding machines.
These objects can be parameterized in the quadratic family in much the same way
as periodic points can be parameterized. Each adding machine is a limit of a
parameterized family of almost adding machines. The almost adding machine
families give a clearer picture of the position of the adding machines in the
quadratic family. This talk will give some applications of this approach.

Dmitry Kleinbock
( Brandeis University )Bounded geodesics in moduli space
This is a joint work with Barak Weiss. In the moduli space of complex structures on a Riemann surface,
we construct a set of full Hausdorff dimension of points
with bounded Teichmuller geodesic trajectories.
The main tool is quantitative nondivergence of
Teichmuller horocycles, due to Minsky and Weiss.
This has an application to billiards in rational polygons.

Janina Kotus
( Warsaw University of Technology )Geometry and Ergodic Theory of nonrecurrent Elliptic Functions
We explore the class of elliptic functions whose all critical points
contained in the Julia set are nonrecurrent and whose $\omega$limit
sets form compact subsets of the complex plane. In particular, this
class comprises hyperbolic, subhyperbolic and parabolic elliptic maps.
Let $h$ be the Hausdorff dimension of the Julia set of such elliptic
function $f$. We construct an atomless $h$conformal measure $m$ and
we show that the $h$dimensional Hausdorff measure of the Julia set of
$f$ vanishes unless the Julia set is equal to the entire complex plane
$C$. The $h$dimensional packing measure is always positive and it is
finite if and only if there are no rationally indifferent periodic points.
Furthermore, we prove the existence of a (unique up to a multiplicative
constant) $\sigma$finite $f$invariant measure $\mu$ equivalent with $m$.
The measure $\mu$ is then proved to be ergodic and conservative and we
identify the set of those points whose all open neighborhoods have
infinite measure $\mu$. In particular we show that $\infty$ is not
among them.
( joint work with Mariusz Urbanski)

Krystyna Kuperberg
( Auburn University )Local modifications of continuous flows
Continuous dynamical systems on manifolds can be
locally modified with plugs in order to achieve the
desired geometry of orbits. Several applications
will be shown. We will also discuss a connection
between the notion of stability in dynamical systems
and the UVproperty in shape theory.

Torsten Lindström
( University of Kalmar, 39182 Kalmar )On the dynamics of discrete foodchains: low and high frequency behavior and optimality of chaos
We analyze a discrete version of Rosenzweig's (Am. Nat. 1973) foodchain model. We provide substantial analytical and numerical evidence for the general dynamical patterns of foodchains predicted by DeFeo and Rinaldi remaining argely unaffected by this discretization. Our theoretical analysis gives rise to a classification of the parameter space into various regions describing distinct governing dynamical behaviors. Predator abundance has a local optimum at the edge of chaos.

Carlangelo Liverani
( University of Rome ``Tor vergata" )Strong statistical properties of Geodesic flows in negative curvature
I will describe a general approach to investigate the statistical
properties of uniformly hyperbolic systems. I will then show that also
some partially hyperbolic systems (Anosov flows) can be treated with
such methods. Finally, I will illustrate some new results that can be
obtained with the above approach for Contact Anosov flows.

Mikhail Lyubich
( Stony Brook )Renormalization in the Henon Family
Renormalization theory of quadraticlike maps is now well understood,
so it is natural to take a look at its higher dimensional relatives.
The Henon family is the simplest twodimensional perturbation
of the quadratic family. In 1980's Collet, Eckmann and Koch suggested
a renormalization scheme that explained the observed parameter
universality for the doubling bifurcations in this family. We will show
that an appropriate scheme justifies some dynamical universalities as
well. However, there are also striking differences between the two
and one dimensional situations which make this subject quite intriguing.
It is an ongoing joint project with Andre de Carvalho and Marco Martens.

John Mayer
( Univ. of Alabama at Birmingham )Indecomposable continua and the Julia sets of polynomials, II
We find necessary and sufficient conditions for
the connected Julia set of a polynomial of degree $d\ge 2$ to be
an indecomposable continuum. One necessary and sufficient
condition is that the impression of some prime end (external ray)
of the unbounded complementary domain of the Julia set has
nonempty interior in $J$. Another is that every prime end has as
its impression the entire Julia set. The latter answers a
question posed in 1993 by the authors.
This is joint work with James T. Rogers, Jr. (Tulane University).

Eugen Mihailescu
( UNT )Dynamics of holomorphic maps
on projective spaces
In this talk I will go over some aspects of dynamics of holomorphic maps
on projective spaces $\mathbb P^k \mathbb C$.
Part of these results are joint work with M. Urbanski.
Given a holomorphic map which is noninvertible one can define several
notions of inverse entropy and inverse topological pressure and then use
them for estimates for the Hausdorff dimension of the intersection between
the stable manifolds and the basic sets of saddle type $\Lambda$. The
situation is very different from the case of diffeomorphisms since now we
have to deal with the possibly infinitely many prehistories that a point
can have in $\Lambda$. This will imply that the notions introduced have
some different properties than the classical (forward) ones.

Michal Misiurewicz
( IUPUI )Bowen`s entropy, Hausdorff dimension and Lipschitz constant
Bowen`s definition of topological entropy for noncompact
sets provides simple means for the proof of the following theorem.
If $X$ is a metric compact space and $f:X\to X$ a Lipschitz continuous
map, then the Hausdorff dimension of $X$ is bounded from below by the
topological entropy of $f$ divided by the logarithm of its Lipschitz
constant.

Lex Oversteegen
( UAB Birmingham )On minimal maps of 2manifolds
We prove that a minimal selfmapping of a closed
2manifold is monotone and nonseparating (i.e. all points have connected nonseparating preimages); we also prove that there are no minimal selfmappings of 2manifolds with boundary. This implies that the only compact 2manifolds which admit minimal maps are torus and
Klein bottle.

Subbiah Parthasarathy
( Regional Engg. College, Tiruchirappalli )On the multipleperioddoubling bifurcation route to chaos in discrete population dynamical models under seasonal migration
The dynamics of singlespecies population with nonoverlapping generation has commonly been analysed by using onedimensional discrete maps, which can display universal single perioddoubling bifurcation route to chaos, during free growth. Migration is a common ecological process which can affect the size of the population. Here we consider the effect of migration (both emigration and immigration) in the two predominantly used discrete population growth models namely, i) Logistic map and ii) Ricker map. In our earlier study, we have shown that the effect of a very simple case of migration, where emigration/depletion is assumed as constant at every generation in Ricker map. We have shown that the Ricker map with constant emigration displays unusual dynamics of extinction, in which the populations can persist under a high rate of emigration/harvesting while even for a lower rate of emigration leads to extinction for the same growth rate. Now we consider a more realistic case in which the effect of seasonally varying migration is included in these two models. We show that these two models under seasonal migration displays a wide variety of complicated bifurcation structures and exhibit a mulitipleperioddoubling bifurcation route to chaos.

Karl Petersen
( University of North Carolina at Chapel Hill )Sideways Symbolic Dynamics
Usually symbolic dynamics studies properties of the shift
transformation on the space of sequences on a finite alphabet. The
adic transformations defined by A. Vershik are in a way transverse to
the shift. Particular examples of adic transformations provide
intriguing examples of dynamical behavior, with implications for the
amount of information retained in the tail fields of probability
theory. We discuss recent results about the dynamics of these systems
along with some open questions on which progress is being made.

Pawel Pilarczyk
( Jagiellonian University (Krakow) and Georgia Tech (Atlanta) )Algorithmic Homology Computation
and the Conley Index
We are going to talk about the recently developed algorithms
which allow one to use the computer more efficiently than ever before
to compute the (relative) homology of cubical sets in $\R^n$
(that is, sets built of cubes with respect to a fixed grid),
as well as the homomorphisms induced in homology
by continuous maps on such sets.
We would like to show an example application
of this powerful computationaltopological tool
to the automatic computation of the Conley index.
This is a joint work with K. Mischaikow and M. Mrozek.

J. Leonel Rocha
( Inst. Sup. Eng. Lisboa )Computing Hausdorff dimensions and escape rates
This is a joint work with J. Sousa Ramos. We consider extensions of the kneading theory of Milnor and Thurston to present a weighted kneading theory, introducing weights in the formal power series. We show that this method allow us to derive techniques to compute explicitly topological entropies, Hausdorff dimensions and escape rates.

Lorenzo Sadun
( University of Texas )Deformations of Substitution Tiling Spaces
We consider substitution tilings in $d$ dimensions and the action of the translation group on these tilings. Deforming the shapes of the tiles, while leaving the combinatorics fixed, creates a new tiling space that is homeomorphic to the original one, but on which the translation group acts differently. (In one dimension such deformations are better known as "time changes").
The basic question is how such deformations affect the dynamics. We construct a map from the space of shape parameters to $H^1(X,R^d)$, where $X$ is the space of tilings. The dynamics depend only on the image of this map. Decomposing $H^1(X,R^d)$ into eigenspaces of the substitution operator, we identify the geometrical significance of deformations in each eigenspace, and give criteria for when two such deformations yield topologically conjugate dynamics.

Petra Sindelarova
( Silesian University in Opava, Czech Republic )Proof of a conjecture on omegalimit sets for a continuous zero toppological entropy map of the unit interval
For a continuous map f of the unit interval I with zero topological entropy it is known that, for any omegalimit set \omega, the set P(\omega)={x \in I; \omega_f(X)=\omega} is of type F_\sigma\delta. At the European Conference on Iteration Theory in Evora, Portugal 2002, A.N. Sharkovsky asked whether
P(\omega) is a G_\delta\sigma set. We will answer this question.

Jaroslav Smital
( Mathematical Institute, Silesian University, Opava, Czech Republic )The classification of triangular maps with zero topological entropy
For continuous maps of the interval there are several tens of equivalent conditions characterizing zero topological entropy. Most of these conditions are not equivalent for triangular maps $F(x,y)=(f(x),g(x,y))$ of the square, even when they are applicable. Based on very recent results obtained jointly with G.L. Forti and L. Paganoni, we can exhibit the complete list of mutual relations between them for triangular maps which are nondecreasing on the fibres. For general triangular maps some relations remain unknown, but again we provide nontrivial examples showing that some implications are not valid.

Bernd O. Stratmann
( University of St Andrews, Scotland )The Geometry of Kleinian Limit Sets
The talk will give a survey on various recent results
concerning the Diophantine, fractal, multifractal and
thermodynamical geometry of limit sets of Kleinian groups.

Jianzhong Su
( University of Texas at Arlington )Dynamics of elliptic bursting
Elliptic bursting arises from fastslow systems and involves
recurrent alternation between active phases of large amplitude oscillations and silent phases of small amplitude oscillations. This talk concerns with a rigorous analysis of elliptic bursting with and without noise. We first prove the existence of elliptic bursting solutions for a class of fastslow systems without noise by establishing an invariant region for the return map of the solutions. For noisy elliptic bursters, the bursting patterns depend on random variations associated with delayed bifurcations. We provide an exact formulation of the duration of delay and analyze its mean and variance. The duration of the delay, and consequently the durations of active and silent phases, is shown to be closely related to the logarithm of a distance function that is nearly Gaussian and propotional to the amplitude of the noise. The treatment of noisy delayed bifurcation here is a general theory of delayed bifurcation valid for other systems involving delayed bifurcation as well, and is a continuation of the rigorous ShishkovaNeishtadt theory on delayed bifurcation or delay of stability loss.

Meiyu Su
( Long Island University, Brooklyn )Laminations for Hyperbolic Measures
We study the lamination structure of a diffeomorphism of a compact smooth Riemannian manifold. It turns out that for every invariant and ergodic measure, if it is hyperbolic there exist dynamically contracting and expanding
laminations continuously injected into the global stable and unstable partitions and filling up the measure. The two laminations are intersecting locally transversely. The new ingredients are the countable collection of the Pesin boxes and the entropy Topology.

Mike Sullivan
( SIU/UNT )Linking of minimal sets
This is joint work with Alex Clark. We define a linking homomorphism between onedimensional minimal sets on a flow in 3 dimensions. We study the types of linking that can occur among suspensions of Sturmian minimal sets of the full 2shift embedded in a "Lorenz flow." Here the linking homomorphism is described by a 2x2 matrix. We find a surprising limitation of the realizable linking matrices.

Ilie Ugarcovici
( Penn State University )On admissible geometric codes for geodesics on the modular surface
Two different methods are available for coding geodesics on surfaces of constant negative curvature: a geometric one (Morse coding) obtained by keeping track of the sides of a fundamental region hit by the geodesic, and an arithmetic one (Artin coding) obtained by lifting the geodesic and coding its endpoints.
We present a sufficient condition for a double infinite sequence of integers to be realizable as the geometric code of a geodesic on the modular surface. Moreover, we prove that this class of admissible sequences constitutes a maximal 1step countable Markov chain in the set of all geometric codes. A lower bound estimate for the topological entropy of the corresponding geodesic subflow is computed.
This is joint work with Svetlana Katok.

Klaudiusz Wojcik
( Auburn University and Jagiellonian University )Discrete Conley index criterion for chaos.
We present a discrete version of topological criterion for detecting chaotic dynamics based on the machinery of isoltaing segments. The criterion for chaos based on isolating segments applies directly to differential equations with periodic forcing. It uses two isolating segments whose exit sets on a certain Poincare section are the same but the fixed point indeces computed for the two neighborhoods are different. Despite the direct applicability to differential equations of the criterion it is still hard work to construct analytically the ncessary isolating segments for concrete differential equations. Therefore it would be very helpful to have a discrete counterpart of the geometric criterion, because this would open the way to computer assisted proofs based on such an analogue. Surprisingly, it is not obvious what a discrete counterpart should be. The aim of this talk is to present a possible analogue based on the discrete Conley index theory. This is joint work with Marian Mrozek.

Maciej P. Wojtkowski
( University of Arizona )Rigidity of some Weyl manifolds
with nonpositive sectional curvature
We provide a list of all locally metric Weyl connections
with nonpositive sectional curvatures
on two types of manifolds, ndimensional tori $\Bbb T^n$
and $\Bbb M^n =\Bbb S^1\times\Bbb S^{n1}$
with the standard conformal structures.
For $\Bbb M^n$ we prove additionally that it carries
no other Weyl connections with nonpositive
sectional curvatures, locally metric or not.
Surprisingly the dynamics of Wflow does not necessarily
reflect the negativity of sectional curvatures.

Christian Wolf
( Wichita State University )MorseSmale diffeomorphisms in the Henon family
We consider a oneparameter family of Henon maps and provide a complete description of the dynamics of these maps. It turns out that all maps within this family are MorseSmale diffeomorphisms. This is a joint work with Sandra Hayes.

LaiSang Young
( Courant Institute )A theroy of rank one attractors, Part I
Abstract: By rank one attractors, I refer to attractors in
ndimensional phase spaces that are strongly dissipative with
a single direction of instability. If we think of the theory of
1D maps as leading naturally to the study of Henon attractors
with small b, then the work I present can be seen as the next
natural stage of development. I will give conditions to guarantee
the existence of SRB measures with strong stochastic properties,
and discuss applications to periodically forced oscillators and
Hopf bifurcations. (This is joint work with Don Wang.)

LaiSang Young
( Courant Institute )A theory of rank one attractors, Part II
This is a continuation of my first talk, the goal
of which is to put things in context. In this second hour,
I will give more detail on the dynamical behavior of the maps
in question, focusing on the following aspects as time permits:
(1) 1D models, (2) global structure of attractor, (3) geometry
of critical set, (4) local analysis  and how it mimicks 1D.

Anna Zdunik
( Warsaw University, Poland )Conformal, invariant and Hausdorff measures for the exponential family.
For a large class of maps $f(z)=a\exp(z)$ we study the
dimension of some natural, dynamically defined essential subset of
the Julia set (It can be understood as an analogue of the conical
limit set). A conformal measure supported on this special set is
built and its ergodic properties are studied. We also introduce
the corresponding thermodynamical formalism and study the
dependence of the above dimension on the parameter $a$. These
results have been obtained jointly with Mariusz Urbanski.

Michel Zinsmeister
( Univ Orleans )Some remarks on laplacian growth
During the past two decades physicists have developped different theories of random growth modelizing more or less turbulent physical phenomena. In particular Hastings and Levitov have introduced a one parameter family of laplacian growths whose simulation exhibits some kind of transition to turbulence. The aim of the talk is an attempt to a mathematical understanding of this phase transition.