Maryland-Penn State Semiannual Workshop
on Dynamical Systems and Related Topics
University of Maryland, March 15-18, 2008


Participants have many more interesting talks than we have talking spaces. We've solicited titles and abstracts for "virtual talks" as below, with the idea that they may publicize some work and alert participants to things they might want to talk about.

Jayadev Athreya ,   Princeton University
Lattice point problems and volume growth for Teichmuller space
      In joint work with Sasha Bufetov, Alex Eskin, and Maryam Mirzakhani, we apply some of the ideas of the Ph.D. Thesis of G. A. Margulis to Teichmuller space. Let x be a point in Teichmuller space, and let B_R(x) be the ball of radius R centered at x (with distances measured in the Teichmuller metric). We obtain asymptotic formulas as R tends to infinity for the volume of B_R(x), and also for for the cardinality of the intersection of B_R(x) with an orbit of the mapping class group.

Vitaly Bergelson ,   Ohio State University
Distribution of values of bounded generalized polynomials
Click here for an abstract. Acta Math., to appear. Joint work with Alexander Leibman.

Lewis Bowen ,   University of Hawaii
A new isomorphism invariant of free group actions
      The invariant of the title can be used to completely classify Bernoulli shifts over a free group up to isomorphism. This answers a question of Ornstein and Weiss. The new invariant is similar in some respects to the classical Kolmogorov-Sinai entropy.

Lewis Bowen ,   University of Hawaii
Invariant measures on the space of horofunctions of a word hyperbolic group
      There is a natural equivalence relation on the space H_0 of horofunctions of a word hyperbolic group that take the value 0 at the identity. The focus of the talk is on the measures that are invariant under this relation. Main results: (1) there are only finitely many such measures that are ergodic. This fact can be viewed as a discrete analog of the Bowen-Marcus theorem. (2) if \eta is such a measure and G acts on a probability space (X,\mu) by measure-preserving transformations then \eta \times \mu is virtually ergodic with respect to a natural equivalence relation on H_0 \times X. This is comparable to a special case of the Howe-Moore theorem. (3) These results are applied to prove a new pointwise ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space. This result is new even in the special case in which G is a free group because there are no restrictions on the choice of generating set.

Maria Isabel Cortez ,   Universidad de Santiago de Chile
Generalized odometers and omega-limit sets which admit infinitely many ergodic measures
      We give conditions in order that the generalized odometer associated to a unimodal map admits infinitely many ergodic measures. Then, we prove that under some restrictions, the sets of invariant probability measures of the associated generalized odometer and the omge-limit set are affine homeomorphic. Joint work with Juan Rivera-Letelier. For the slides of this talk, click here.

Katrin Gelfert ,   Northwestern University
Non-hyperbolic interval maps and their Lyapunov exponents
      The talk concerns aspects of the dynamics of interval maps, and the at all possible behavior of $C^2$ expansive Markov maps. It includes, in particular, properties of the spectrum of Lyapunov exponents as well as geometric properties of emerging fractal sets and level sets of certain exponents. Here I focus on a characterization by means of their Hausdorff dimension. The analysis covers several classes of non-hyperbolic maps, including maps with parabolic periodic points. (This material concerns a crossection of the two papers with M. Rams (IMPAN, Warsaw): Geometry of limit sets for expansive Markov systems, to appear Trans AMS, and The Lyapunov spectrum of some parabolic systems, submitted. See also other work on my home page.

Anton Gorodetski ,   University of California, Irvine
Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes
      We prove that for a generic $C^1$-diffeomorphism existence of a homoclinic class with periodic saddles of different indices (dimension of the unstable bundle) implies existence an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of $f$. The preprint can be found by clicking here. (This is a joint work with Lorenzo Diaz)

Inger Knutson ,   University of Agder
Weak mixing implies weak mixing of higher orders along tempered functions
      (Joint work with Vitaly Bergelson) We extend the weakly mixing PET (polynomial ergodic theorem) to much wider families of functions. Besides throwing new light on the question of "how much higher degree mixing is hidden in weak mixing", the obtained results also show the way to possible new extensions of the polynomial Szemeredi theorem.

Mahesh Nerurkar ,   Rutgers University - Camden
Density of certain SL(2,R)-valued cocycles satisfying a Liouville type condition.
      Consider the class of $C^r$-smooth $SL(2,R)$ valued cocycles, based on the rotation flow on two torus with irrational rotation number $\alpha$. We show that in this class, cocycles with positive Lyapunov exponents are dense if $\alpha$ satisfies the following Liouville type condition: $\big| \alpha - \frac {p_n}{q_n}\big| \leq C \text {exp}(-q^{r+1+\delta}_n)$, where $C>0$ and $\delta>0$ are some constants and $\frac {P_n}{q_n}$ is some sequence of irreducible fractions.

Mark Pollicott ,   University of Warwick
Points where conjugacies are not differentiable
      Consider two C^2 expanding maps S and T on the circle which are topologically conjugate, but not C^1 conjugate. It is easy to see that the conjugating map must have zero derivative almost everywhere (i.e., a "slippery Devil's staircase"). In fact the complement X of the set where derivative is zero has Hausdorff dimension strictly between 0 and 1, and this value can be evaluated using pressure and theormodynamic formalism. For example, if S(x) = 2x (mod 1) and T(x) = 2x + (1/7) Sin(2\pi x) (mod 1) then the conjugating map has zero derivative, except on a set of Hausdorff dimension 0.9926... (This is joint work with Thomas, Jordan)

Ian Putnam ,   University of Victoria
Orbit equivalence for minimal actions of finitely generated abelian groups on a Cantor set
      In earlier work with T. Giordano and C. Skau, we provided a complete invariant for orbit equivalence for minimal AF-relations and minimal actions of the group of integers on the Cantor set. With the same co-authors and H. Matui, we have extended this classification, first, to include minimal Cantor actions of the free abelian group on two generators (J. AMS, to appear; for the preprint click here) and, more recently, minimal Cantor actions of any finitely generated abelian group.

Ian Putnam ,   University of Victoria
A homology theory for basic sets
      We consider Smale spaces, a particular class of hyperbolic topological dynamical systems, which include the basic sets for Smale's Axiom A systems. We present a homology theory for such systems which is based on the dimension group in the special case of shifts of finite type. This theory provides a Lefschetz formula relating trace data with the number of periodic points of the system. (A brief survey may be found by clicking here .)

Nandor Simanyi ,   University of Alabama at Birmingham
Unconditional Proof of the Boltzmann-Sinai Ergodic Hypothesis
      We consider the system of n (n>1) elastically colliding hard balls of masses m_1, m_2, ... , m_n and radius r on the flat unit torus T^d, d>1. The goal is to sketch the main dynamical and geometric ideas behind the proof of the so called Boltzmann-Sinai Ergodic Hypothesis for such hard ball systems, i. e. the full hyperbolicity and ergodicity for every selection (m_1,..., m_n; r) of the external geometric parameters, without the assumption that almost every singular trajectory is geometrically hyperbolic (sufficient), i. e. that the so called Chernov-Sinai Ansatz holds true for the model. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical and geometric methods. The core of the proof is a newly developed, thorough geometric analysis of the relation between the singularity manifolds and the submanifolds describing the trajectories (more precisely, the trajectory segments) that are non-sufficient, despite the combinatorial richness of their symbolic collision sequence.

Anatoly Vershik ,   Steklov Institute, Saint Petersburg
Scaling entropy and random walks
      The year 2008 is a distinguished year for specialists in Dynamical Systems not only because Misha's Brin 60-th (however this is the main event) but also because this is 50 years after the invention of Kolmogorov entropy. I suggested a new notion which joins together Kolmogorov-Sinai entropy and Kolmogorov-... epsilon- entropy. Up to now this new notion (so called "Scaling entropy") is well understood not for automorphisms but for filtrations (=decreasing sequences of sigma-fields). In the paper Scaled entropy of filtrations of sigma-fields with my student A.Gorbulsky we applied this notion in order to distinguish random walks in a random group like environment.

Hong Kun Zhang ,   Northwestern University
Decay of correlations for nonuniformly hyperbolic billiards
      Billiards are dynamical systems originating from the study of Boltzmann's Ergodic Hypothesis in statistical mechanics. For nonuniformly hyperbolic billiards, it is very important to determine the rate of convergence of sequence of distributions to equilibrium state. We introduce a new one-step expansion condition, and show the reduced map of billiard map on a uniformly hyperbolic set enjoys exponential decay of correlations. The main method we use is to couple distributions supported on two families of unstable curves, and show the couple rate is exponentially fast. This is a joint work with Nikolai Chernov.

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