(September 10) Randall McCutcheon: Some Ramsey-theoretic results for amenable groups - The theorems of Schur, van der Waerden, and Roth are three of the better known results of Ramsey theory. The first two belong to partition Ramsey theory, while Roth's theorem belongs to density Ramsey theory. This talk, which concerns joint work with V. Bergelson and Q. Zhang, focuses on extensions of these three results (for van der Waerden's theorem, the extension is limited to the case of length four progressions) in amenable groups.

(September 17) Joe Auslander: Equivalence relations and the capturing property in topological dynamics - Let $(X,T)$ be a flow (a jointly continuous action of the group $T$ on the compact Hausdorff space $X$.) If $K\subset X$ the capturing set $C(K)$ is the set of $x\in X$ whose orbit closure meets $K$. If we consider the product action of $T$ on $X\times X$, then $C(\Delta)=P$, the proximal relation. We obtain an increasing family of \lq\lq proximality" relations by alternating the closure and capturing relations: $P_1=P$, and inductively $P_ {\gamma +1}=C(\overline {P_\gamma})$. (If $\gamma$ is a limit ordinal define $P_{\gamma}=\cup [P_i|i<\gamma]$.) If the flow $(X,T)$ is minimal, this procedure terminates in at most countably many steps, and provides a characterization of the distal structure relation. A characterization of the equicontinuous structure relation is also obtained.

(October 15) Joseph Previte: Topological Dynamics on Moduli Spaces - Let M be a one-holed torus with boundary B (a circle) and G the mapping class group of M fixing B. The group G acts on the space S of SU-gauge equivalence classes of flat SU-connections on M with fixed holonomy on B. We study the topological dynamics of the G-action and give conditions for the individual G-orbits to be dense in S. A strategy for extending this result to arbitrary surfaces M will also be presented.

(October 22) Michael Brin: On the ergodicity of the geodesic flow on surfaces of nonpositive curvature. - Let S be a compact (nonflat) surface of nonpositive curvature and let A be the (open) set of points where the curvature is negative. It is known that the geodesic flow on S is ergodic if almost every geodesic intersects A.

Theorem. If A has finitely many connected components, then almost every geodesic intersects A (and consequently, the geodesic flow is ergodic).

(October 29) Howie Weiss : Some New Results About Continued Fractions Via Multifractal Analysis of the Gauss Map - In this talk we extend some of the theory of multifractal analysis for conformal expanding systems to two new cases: the non-uniformly hyperbolic example of the Manneville-Pomeau equation, and the continued fraction transformation. A common point in the analysis is the use of thermodynamic formalism for transformations with infinitely many branches.

We apply the multifractal analysis to prove some new results on the precise exponential speed of convergence of the continued fraction algorithm. This gives new quantitative information on geodesic excursions up cusps on the modular surface.

(November 5) Jun Hu: The Julia set of the Feigenbaum quadratic polynomial - Julia sets, introduced for complex analytic maps, supply a lot of examples of fractal sets which can be understood. The idea of renormalization developed a way to study the geometric properties of certain nonhyperbolic systems. The Feigenbaum quadratic polynomial, which is an accumulation of period-doubling bifurcation in the logistic family of quadratic polynomials, is infinitely renormalizable. Its real dynamics has a Cantor set attractor with some geometric similarity inside, and its complex dynamics has corresponding geometric properties for the Julia set and then it implies some global regularities of this fractal set, such as local connectivity and etc.. In this talk, we will try to explain these aspects.

(November 12) Joe Rosenblatt: Convergence of Convolution Powers - Spectral conditions on a probability measure on $mathbb Z$ were known to guarantee good behavior of the convolution powers as averaging operators. Now these spectral conditions are known to be necessary. These results, as well as generalizations to other groups, give real insight into convergence questions in the ergodic theory of group actions.

(November 13) Rob Benedetto: Dynamics of p-adic Rational Maps - Given an algebraically closed field K and a rational function f(z) in K(z), f may be viewed as a map from the projective line P^1(K) to itself. We consider the resulting discrete dynamical system, that is the study of the action of the set of iterates of f on the projective line. In particular, if K is complete with respect to some absolute value, then we may define the Fatou and Julia sets to be the regions of stability and of chaos, respectively.

The theory is fairly well-developed in the case K = C (the complex field). For example, f maps components of the Fatou set to other components; and according to Sullivan's 1985 No Wandering Domains Theorem, all Fatou components are pre-periodic under this action of f. In this talk we will focus on the case that K is a p-adic field. We will compare and contrast this case with the complex case, develop notions of "components" for the p-adics, and discuss results analogous to Sullivan's Theorem.

(November 19) Mark Levi: Geometry and physics of averaging with applications - I will describe an unexpected appearance of curvature which arises in averaging high frequency vibrations. The geometric observation gives a new insight into several problems such as the mechanism of the Paul trap (which will be described in the talk), stability of the inverted pendula with a vibrating support, composition of non-commuting symplectic matrices and more.

(January 28) Michael Brin: Ergodicity of the geodesic flow: a complete proof. - An attempt will be made to present a complete argument (including the Holder continuity of the spaces of stable Jacobi fields and the absolute continuity of the horospheric foliations) in one hour. A couple of new ideas have reduced the length of the classical proof by Anosov and Sinai.

(February 4) Elon Lindenstrauss: Mean dimension and some applications - Abstract: Mean dimension is a new dimension like invariant for dynamical systems, suggested by M. Gromov. I will describe this invariant and give two applications where this invariant helps to shed light on problems which have been open for some time, whose statement does not involve mean dimension in any way.

(February 25) Chris Hoffman: Rational maps are one sided Bernoulli - Let $f(z)=p(z)/q(z)$ be a rational map of the Riemann sphere, $\bar C$. Freire, Lopes and Ma\~{n}\'{e} proved that for any rational map $f$ there exists a natural invariant measure $\mu_f$. We show that $(\bar C,f, \mu_f)$ is conjugate to the one sided Bernoulli $d$-shift. This is joint work with Deborah Heicklen.

(March 18) Doug Lind: Homoclinic Points and Markov Partitions for Algebraic Z^d-Actions - The first use of Markov partitions to represent toral automorphisms symbolically goes back to Ken Berg's thesis in 1967. The vertices of the parallelograms in such partitions are asymptotic to zero for both large negative as well as positive powers of the automorphism, i.e. are homoclinic points. Recently Schmidt and I have developed an analysis of homoclinic points for the joint action of several commuting group automorphisms. This has been used by Einsiedler and Schmidt to formulate a general approach to constructing Markov partitions for such actions. When applied to a single toral automorphisms it provides a more canonical method that that of Vershik and Kenyon. However, although this approach is successful in a few cases of joint actions, its generality is not understood.

(April 1) Mike Boyle : Expansive invertible onesided cellular automata - (joint work with A. Maass) Suppose f is an expansive homeomorphism commuting with the onesided full shift on N symbols (i.e. f is an invertible expansive onesided cellular automaton) and f is sofic. We prove that for some integer J, divisible by the same primes as N, f must be a mixing SFT shift equivalent to a full shift on J symbols, and if N is a power of a prime p, then N \geq p^2. Our tools are certain dimension groups and measures on unstable and stable sets. Combinatorial constructions in special cases provide further evidence for some conjectures. (A preprint is on my home page.)

(April 15) Yasha Pesin: Dimension and product structure of hyperbolic measures - I will discuss the long-standing Eckmann--Ruelle conjecture in dimension theory of smooth dynamical systems: the pointwise dimension of every hyperbolic measure invariant under a $C^{1+\alpha}$ diffeomorphism exists almost everywhere. This implies the crucial fact that virtually all the characteristics of dimension type of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide.

The proof of the conjecture is based on the fact that every hyperbolic invariant measure possesses asymptotically ``almost'' local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. This has not been known even for measures supported on locally maximal hyperbolic sets.

(April 29) Brian Hunt : Optimal Orbits of Hyperbolic Systems - Given a smooth dynamical system on a compact Riemannian manifold M, and a Lipschitz function F : M -> R, consider the question of which invariant measure(s) maximize the average of F. I will discuss past and present results related to the conjecture that for generic F, the maximum is achieved by a measure supported on a periodic orbit. In particular I will present joint work with Guo Cheng Yuan for the case of hyperbolic dynamical systems.