on Dynamical Systems and Related Topics

University of Maryland, March 19-22, 2005

**
Jose Alves**, University of Porto
*
Smooth conjugacies of interval maps
*

We consider a topological conjugacy between
interval maps. We show that for certain maps
if the conjugacy is differentiable at some point,
then it is differentiable in an open interval.

**
Sarah Bailey
**, University of North Carolina at Chapel Hill

PhD student, advised by
Karl Petersen

*
Computing dimension groups for a certain family of
non-simple Bratteli diagrams
*

Dimension groups of simple Bratteli diagrams are known to be order
isomorphic to the continuous functions modulo the coboundaries. I will
discuss a special family of non-simple Bratteli diagrams for which the
dimension groups are computable. In this family of Bratteli diagrams the
adic transformations are not continuous, but the dimension groups are
still order isomorphic to the continuous functions modulo the continuous
coboundaries.

**
Youngna Choi,
Montclair State University
**,
*
Topology of Attractors from Two-Piece Expanding Maps
*

In this paper we study the topology of the invariant set L_f derived
from an expanding map with one discontinuity f. We classify the
periodic
orbits
on the boundary of L_f that enable L_f to have a trapping
region
and that
do not. We also show that when there are two eventually periodic
orbits on
the
boundary of L_f, the components of L_f form a Markov
partition, and
form
its transition matrix we can calculate the precise entropy of the map.
Furthermore, we show that the set of attractors derived from two piece
expanding
maps is open and dense in the set of all invariant sets derived by the maps.

**
Alex Clark
**, University of North Texas
*
Coding expansive automorphisms of groups and Rauzy fractals
*

Reporting on joint work with Robbert Fokkink,
we will present recent results on
the coding of expansive automorphisms of compact groups.
Making use of the
(non)-archimedean evaluations at the roots of an associated
polynomial, we will
draw connections with generalized Rauzy fractals.

**
Manfred Einsiedler
** Universitaet Wien, Princeton University
*
Measure Rigidity for Cartan actions
*

In an ongoing joint work with Elon Lindenstrauss we describe
invariant measures with positive entropy for the maximal split
torus on a locally homogeneous space defined by a reductive
linear algebraic group (also known as Cartan actions or
Weyl chamber flows). This generalizes earlier work of Katok,
Lindenstrauss and myself regarding SL(n,R)/Gamma to
not necessarily split and not necessarily real groups.
Moreover, if the measure is not algebraic we give restrictions
on the support of the measure.

**
Nikos Frantzikinakis**,
Penn State University
*
Sets of multiple recurrence and convergence
*

For every positive integer k we will construct a set of k-recurrence
but not (k+1)-recurrence. This extends a result of Furstenberg who
produced a set of 1-recurrence but not 2-recurrence. We will also
discuss a similar result for convergence of multiple ergodic averages
and point out a combinatorial consequence related to Szemeredi's theorem
on arithmetic progressions.

This is joint work with E. Lesigne and
M. Wierdl.

**
Daniel Genin
**,
Penn State University

PhD student, advised by
Serge Tabachnikov

*
Hyperbolic outer billiards
*

Mathematical billiards are an extensively studied and in many
respects well understood dynamical systems. For instance, the
question of chaotic behavior was very well explored since Sinai
described the first class of hyperbolic billiards in 1970. Outer
billiards are a much less well known and understood class of
dynamical systems with interesting connections to ordinary
billiards and impact oscillators. After a brief survey of chaotic
ordinary billiards I will describe the first example of a
chaotic outer billiard.

**
Huyi Hu
**, Michigan State University
*
Convergence rates of the transfer operators for sigma finite measures
*

Consider piecewise expanding maps on the unit interval
with indifferent fixed point. It is well known that the
convergence rates to equilibruim under the transfer
operator is polynomial if the systems admit absolutely
continuous probability measure. We study the case
that the invariant measure is sigma finite, and give
both upper and lower bound estimates for the convergence
rates.

This is a joint work with Nicolai Haydn.

**
Anatole Katok
**,
Penn State University
*
New geometric method in differentiable rigidity of
partially hyperbolic actions
*

I will explain the main ingredients of the proof of local
differentiable rigidity (up to a finite-dimensional family of
standard algebraic perturbations) for
restrictions of the Weyl chamber flow on SL(n,R) to a plane in
general position. The method of proof
is radically different and complementary to the KAM iteration scheme
which has been successfully applied to actions by partially hyperbolic
automorphisms of the torus. It uses, among other things,
ideas and results from algebraic K-theory. This is a part of an
ongoing joint work with Danijela Damjanovic.

**
Elon Lindenstrauss
**, Princeton University
*
On measures invariant under tori and other actions
*

(Math department colloquium, Friday March 18)

An important open problem in the theory of flows on
locally homogeneous spaces is the classification of
measures invariant under actions of multidimensional
diagonalizable groups. A similar classification due
to Ratner for measures invariant under actions of
unipotent groups has had numerous applications in
many areas, and already the partial results we have
towards measure classification for diagonalizable
actions have found some applications in number theory
and arithmetic quantum chaos.

I will survey some recent results regarding these invariant
measures and their applications.

**
Hee Oh
**, Cal Tech
*
Equidistribution of solvable flows
*

We will explain the equidistribution result of certain solvable groups
on $G/\Gamma$ (as usual, $G$ is semisimple and $\Gamma$ a lattice).
More precisely, if $H$ is a semisimple subgroup of $G$ with $H=KAN$
an Iwasawa decomposition, and the orbit $Hx$ is dense in $G/\Gamma$,
then $Qx$ is equidistributed in $G/\Gamma$ for any closed subgroup $Q$ of
$H$ containing $AN$.
As an application, we obtain the equidistribution of lattice orbits on the
Furstenberg boundary of $G$. Moreover we show that this equidistribution
on the Furstenberg boundary is independent from the distribution of
lattice orbits in the symmetric space. This is a joint work with Alex
Gorodnik.

**
Jens Marklof **, Bristol

*
Limit theorems for skew translations
*

Strongly chaotic systems such as Anosov maps are well known to satisfy
central theorems for the distribution of long trajectories. The case of
ergodic, but not mixing, skew translations seems more subtle, and I will
discuss some of the recent developments and open conjectures.

**
Kyewon Park
**, Ajou University
*
Complexity of entropy zero dynamical
systems
*

Motivated by the study of actions of Z^2 and more general groups, and
their subgroup actions, we investigate entropy type invariants for deterministic
systems. Several refinements of the notion of entropy have been introduced by
various authors. We propose a new notion, the entropy dimension and develop the
basic definitions. We consider their behaviors on examples in the case of Z
actions and Z^2 actions.

**
Karl Petersen
**, University of North Carolina at Chapel Hill
*
The adic transformation on the Euler graph
*

The Euler graph has vertices labeled (n,k) for
n=0,1,2,... and k=0,1,...,n, with k+1 edges from (n,k) to (n+1,k) and
n-k+1 edges from (n,k) to (n+1,k+1). The number of paths from (0,0) to
(n,k) is the Eulerian number A(n,k), which is the number of permutations
of 1,2,...,n+1 with exactly n-k falls and k rises. The symmetric measure
is defined by assigning equal weights 1/(n+2) to all of the edges downward
from each level n. We prove that the adic transformation on the Euler
graph is ergodic with respect to the symmetric measure. The argument
adapts a previous proof by Keane of ergodicity of the symmetric Bernoulli
measure for the Pascal adic by analyzing the relevant random walk. A
study of the asymptotic properties of the Eulerian numbers yields the
stronger result that in fact the symmetric measure is the unique fully
supported ergodic invariant Borel probability measure for this system.
These results are part of an ongoing joint project with Sarah Bailey,
Michael Keane, and Ibrahim Salama in various combinations.

**
Mark Pollicott**,
University of Warwick, England
*
Dynamical zeta functions and Dilation Equations
*

Dynamical zeta functions (in the sense of Ruelle) often provide a
very useful tool for studying characteristics associated to
hyperbolic dynamical systems (e.g., Lyapunov exponents, Hausdorff
Dimension, etc.). In particular, this can often lead to an
effective method for their computation. We want to apply this
approach to the study of the regularity of solutions to Dilation
Equations f(x) = \sum_k c_k f(2x - k) on the real line, developing
an idea of Cohen and Dauberchies. This has applications to the
theory of Wavelets, and is joint work with Howie Weiss.

**
Mrinal Roychowdhury,
** Wesleyan University

PhD student, advised by
Mike Keane

*
Finitary orbit equivalence
*

Two invertible dynamical systems $(X, \mathfrak{A}, \mu, T)$ and $(Y,
\mathfrak{B}, \nu, S)$ are said to be orbit equivalent if there exists an
invertible measure-preserving mapping $\phi$ from a subset of $X$ of
measure one to a subset of $Y$ of full measure such that
$\phi(\text{Orb}_T(x))=\text{Orb}_S\phi(x)$ for $\mu$-almost every $x\in
X$. Suppose that the above $X$ and $Y$ are also topological spaces and
that $T$ and $S$ are homeomorphisms. We say that the orbit equivalence is
finitary if both $\phi$ and $\phi^{-1}$ are continuous after restriction
to sets of measure 1. I will discuss how an irrational rotation and the
binary odometer are finitarily orbit equivalent.

**
Daniel J. Rudolph
**, Colorado State University
*
Mixing, disjoint factors and isometric extensions
*

Recently Francois Parreau and Jean-Paul Thouvenot
have proven the rather startling fact that any ergodic transformation
that is not mixing has a nontrivial factor disjoint from all mixing actions.
This result has a relative version and with it one is able to prove
a relative version of the fact that weakly mixing isometric extensions
of mixing actions must be mixing. This can then be used to show
that for discrete amenable group actions, weakly mixing isometric
extensions of mixing actions must be mixing, using the orbit transference
method.

My goal is to sketch the machinery of the Parreau-Thouvenot argument
and how it can used to carry out this program.

**
Omri Sarig**,
Penn State University
*
Invariant Radon measures for the horocycle flow on periodic
surfaces
*

This is joint work with Francois Ledrappier.

The horocycle flow of a non-geometrically finite hyperbolic
surface may have many different (infinite) ergodic invariant Radon
measures. (This should be contrasted with co-compact or co-finite volume
case where there is just one non-trivial measure - up to scaling).

We classify these measures for the class of periodic surfaces: regular
covers of surfaces of finite volume.
It turns out that in this case there are as many measures as there are
positive eigenfunctions for the Laplacian of the surface.

**
Nandor Simanyi
**, University of Alabama at Birmingham
*
Rotation sets of billiards
*

So far we have studied the rotation sets of two families of
billiards:

(A) The inertia motion of a point in the flat torus $\Bbb T^m$ minus a strictly
convex, compact obstacle $O$ with a smooth boundary. Here, by definition,
the rotation set $R$ consists of all limiting points of the average
displacements along orbit segments, when the lengths of these segments tend
to infinity.

(B) The inertia motion of a point in a rectangle minus a strictly convex,
compact obstacle $O$ with a smooth boundary inside the interior of the
rectangle. Here the rotation set $R$ is, by definition, the set of all limiting
values of the average wrapping around the obstacle by orbit segments whose
lengths tend to infinity.

We will present a detailed description of the above rotation sets. The proofs
use ideas from geometry, topology, and a bit of combinatorics.

(These are joint results with A. Blokh and M. Misiurewicz.)

**
Howard Weiss**,
Penn State University
*
How smooth is your wavelet: a thermodynamic formalism approach
*

The study of wavelet regularity is currently a major topic of
investigation having important applications. In this talk I will discuss our
unified approach to wavelet regularity via thermodynamic formalism.
The first part of the talk will be a quick introduction to wavelets.

This is joint work with Mark Pollicott.

**
Tom Ward
**, University of East Anglia
*
Disjointness and entropy geometry
*

This is joint work with Manfred
Einsiedler, in which the entropy geometry
of commuting automorphisms of zero-dimensional
groups is used to show that many natural
actions are mutually disjoint.

**
Tamar Ziegler
**, Ohio State University
*
Configurations in sets of positive upper
density in R^m
*

We apply ergodic theoretic tools to solve a classical problem in
geometric Ramsey theory: let E be a measurable subset of R^m, with
positive upper density. Let V={0,v_1,...,v_k} be a subset of R^m. We show
that for r large enough, we can find an isometric copy of rV arbitrarily
close to E. This is a generalization of the work of Furstenberg,
Katznelson and Weiss proving this property for k=m=2.