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.

Kneading theory for two-dimensional skew-product maps

This is a joint work with J. Sousa Ramos. We consider skew-product 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 one-dimensional 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 two-dimensional maps. During this work we only consider periodic and preperiodic points. Some examples are provided (Baker map, Kaplan-Yorke map, etc.).

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.

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.

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.

Attractors for graph-critical 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 non-recurrent 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.

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 time-periodic flows of Stokes and Euler type using a combination of topological methods from the Thurston-Nielsen theory and analytical methods from the stability theory of periodic linear ODE's.

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.

New exact non-symmetric 3D MHD equilibria

In this talk I will present new methods of constructing exact three-dimensional 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.

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 Sze-Man 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.

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

Orbits of unbounded energy in generic quasi-periodic perturbations of geodesic flows of certain manifolds

We show that certain mechanical systems, including geodesic flow plus a quasi-periodic 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 non-degeneracy 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

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.

Topological invariants in forced piecewise-linear Fitzhugh-Nagumo-like systems

This is a joint work with J. Sousa Ramos. Mathematical models for periodically-forced excitable systems arise in many biological and physiological contexts. In previous papers, chaotic dynamics of a forced piecewise-linear Fitzhugh-Nagumo-like 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.

On Traveling Wave Solutions and Proper Solutions To the Two-Dimensional Burgers-Korteweg-de Vries Equation

In this paper, the stability of a two-dimensional autonomous system is analyzed, which indicates that under some particular conditions, the two-dimensional Burgers-Korteweg-de Vries (2D-BKdV) equation has a unique bounded traveling wave solution. Then by using the qualitative theory of differential equations, the traveling solitary wave solution to the 2D-BKdV equation is established. At the end of the paper, the asymptotic behavior of the proper solutions of the 2D-BKdV equation is presented.

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

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 Perez-Marco 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.

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 bang-bang 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

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.

The Dynamics of a Taffy-Pulling 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 taffy-pulling 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.

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.

The Distribution of the Measure of Cylindersets for Non-Gibbsian Measures

For ergodic measures the theorem of Shannon-McMillan-Breimann 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 re-entry times.

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.

Rigidity of Low Dimensional Group Action

rigidity of infinite volume 3-manifolds with sectional curvature $-b^2\le K\le -1$ and generalize the rigidity of Hamenstädt or more recently Besson-Courtois-Gallot, to 3-manifolds with infinite volume and geometrically infinite fundamental group.

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.

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 one-parameter family of real cubic polynomials with negative Schwarzian derivatives.

A classification of inverse limit spaces of tent maps with periodic critical points

We work within the one-parameter 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.

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.

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 non-compact 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 non-compact case an iterated system may have more than one limit. \end{abstract} \end{document}

Adding Machines in the Quadratic Family of Functions

Authors: James Keesling and Louis Block It is well-known 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.

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.

Geometry and Ergodic Theory of non-recurrent Elliptic Functions

We explore the class of elliptic functions whose all critical points contained in the Julia set are non-recurrent 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)

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 UV-property in shape theory.

On the dynamics of discrete food-chains: low- and high frequency behavior and optimality of chaos

We analyze a discrete version of Rosenzweig's (Am. Nat. 1973) food-chain model. We provide substantial analytical and numerical evidence for the general dynamical patterns of food-chains 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.

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.

Renormalization in the Henon Family

Renormalization theory of quadratic-like maps is now well understood, so it is natural to take a look at its higher dimensional relatives. The Henon family is the simplest two-dimensional 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.

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).

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 non-invertible 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.

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.

On minimal maps of 2-manifolds

We prove that a minimal self-mapping of a closed 2-manifold is monotone and non-separating (i.e. all points have connected non-separating preimages); we also prove that there are no minimal self-mappings of 2-manifolds with boundary. This implies that the only compact 2-manifolds which admit minimal maps are torus and Klein bottle.

On the multiple-period-doubling bifurcation route to chaos in discrete population dynamical models under seasonal migration

The dynamics of single-species population with non-overlapping generation has commonly been analysed by using one-dimensional discrete maps, which can display universal single period-doubling 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 mulitiple-period-doubling bifurcation route to chaos.

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.

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 computational-topological tool to the automatic computation of the Conley index. This is a joint work with K. Mischaikow and M. Mrozek.

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.

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.

Proof of a conjecture on omega-limit 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 omega-limit 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.

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.

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.

Dynamics of elliptic bursting

Elliptic bursting arises from fast-slow 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 fast-slow 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 Shishkova-Neishtadt theory on delayed bifurcation or delay of stability loss.

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.

Linking of minimal sets

This is joint work with Alex Clark. We define a linking homomorphism between one-dimensional 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 2-shift 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.

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 1-step 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.

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.

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, n-dimensional tori $\Bbb T^n$ and $\Bbb M^n =\Bbb S^1\times\Bbb S^{n-1}$ 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 W-flow does not necessarily reflect the negativity of sectional curvatures.

Morse-Smale diffeomorphisms in the Henon family

We consider a one-parameter 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 Morse-Smale diffeomorphisms. This is a joint work with Sandra Hayes.

A theroy of rank one attractors, Part I

Abstract: By rank one attractors, I refer to attractors in n-dimensional 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.)

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.

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.

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.