on Dynamical Systems and Related Topics

University of Maryland, March 15-18, 2008

**
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.