# TITLES AND ABSTRACTS OF TALKS

Mahsa Allahbakhshi,   University of Victoria
PhD student, advised by Anthony Quas
Factor triples and relative entropy

Tim Austin,   UCLA
PhD student, advised by Terry Tao
Some recent approaches to multiple recurrence.
Furstenberg's 1977 proof of Szemeredi's Theorem using a conversion to an assertion of multiple recurrence' for probability-preserving systems gave rise to a whole subdiscipline of ergodic theory called Ergodic Ramsey Theory', which went on to provide proofs for several other extremal results in different combinatorial settings. In recent years, a number of alternatives to the original basic strategies of Ergodic Ramsey Theory have emerged within ergodic theory, and offered a clearer insight into the connections between this field and purely combinatorial approaches to the same results. In this talk I will describe Tao's infinitary version of the hypergraph removal lemma from extremal combinatorics, and then sketch how its conjunction with variants of the machinery of pleasant extensions gives a new approach to both the multidimensional multiple recurrence phenomenon and Furstenberg and Katznelson's density version of Hales-Jewett Theorem.

Francesco Cellarosi,   Princeton University
PhD student, advised by Yakov Sinai
On the limit curlicue process for theta sums
I shall discuss a random process achieved as the limit for the ensemble of curves generated by interpolating the values of theta sums. The existence and the properties of this process are established by means of purely dynamical tools and rely on generalizations of a result by Marklof and Jurkat and van Horne. (joint work with Jens Marklof).

Vaughn Climenhaga,   The Pennsylvania State University
PhD student, advised by Yakov Pesin
Multifractal analysis of Birkhoff averages
One important goal in understanding a dynamical system is to describe the asymptotic behaviour of time averages of an observable function---the Birkhoff averages. Given an ergodic invariant measure for the system, the Birkhoff ergodic theorem gives us information on a full measure set of "typical" points. However, there are reasons to want information about the asymptotics at every point, not just almost every point. We mention some of these reasons and describe the key tools of multifractal analysis, which uses dimension theory and thermodynamic formalism to give a more detailed picture.

Van Cyr,   The Pennsylvania State University
PhD student, advised by Omri Sarig
Conformal measures for transient Markov shifts

Andres del Junco ,   University of Toronto
A survey of almost continuous orbit equivalence
Suppose T and S are finite measure-preserving homeomorphisms of topological spaces. By Dye's theorem they are measurably orbit equivalent. If the map implementing the orbit equivalence is continuous when restricted to some set of full measure one could call it almost continuous. It turns out that this definition is too weak but appropriate strengthenings give rise to an interesting theory. We will see that there is an almost continuous version of Dye's theorem, as well as some results for non-singular homeomorphisms and analogues of other theorems in the measurable theory. In particular I will say something about the very interesting notion of almost continuous Kakutani equivalence. Dan Rudolph was quite involved in this field in his last few years.

Matt Foreman,   U.C. Irvine
The classification problem for ergodic measure preserving transformations
In 1932 von Neumann formulated the problem of classifying ergodic measure preserving transformations of [0,1]. In the nearly 80 years that have followed, there has been considerable positive progress on this problem; notably Ornstein's work using the Sinai-Kolmogorov invariant "entropy" to classify Bernoulli shifts and the theorem of von Neumann and Halmos classifying discrete spectrum transformations. After reviewing the history, this talk will focus on joint work with Dan Rudolph and Benjamin Weiss that established some sweeping anti-classification" results. These theorems show that there are fundamental reasons that any classification is impossible.

Hillel Furstenberg ,   Hebrew University
On Meiri Sequences
David Meiri in his Master's thesis inaugurated a study of what he called "generalized correlation sequences". These are sequences whose averages appear in so-called non-conventional ergodic theorems. They are also closely related to what Host and Kra call "Nil-Bohr sets", and they deserve to be analyzed on their own right.

Mike Hochman ,     Princeton University
Geometric rigidity of times-m invariant measures
In 1990 Rudolph showed that if a measure on [0,1] is invariant under times-2 and times-3 mod 1, ergodic, and has positive entropy under the individual maps, then it is Lebesgue measure, settling the positive-entropy case of Furstenberg's "times-2 times-3 conjecture". In particular, this says that if mu is a times-2 ergodic measure of positive entropy, and nu is times-3 ergodic and of positive entropy, and neither is Lebesgue, then mu and nu are not equal. From a result of Host, it follows that in fact mu,nu must be mutually singular. In this talk I will discuss a geometric generalization of this result, showing that it is robust under smooth perturbations: that is, under the same assumptions, if mu',nu' are obtained by applying (different) smooth maps to mu and nu, respectively, then mu',nu' are still singular, and one can identify which is which from the spectrum of a certain flow associated to the measures.

Chris Hoffman ,   University of Washington
A K but not Bernoulli Z^2 subshift of finite type
Domino tilings of the plane have been studied extensively for both their statistical properties and their dynamical properties. In particular Rick Kenyon has shown that the certain properties of a domino tiling are conformally invariant. We construct a subshift of finite type using a collection of colored dominoes. Using these results about domino tilings we will show that our subshift is K (every factor has positive entropy) but is not isomorphic to a Bernoulli shift.

Aimee Johnson ,     Swarthmore College
Recurrence on Infinite Measure Dynamical Systems
Here we consider Z^2 actions on infinite measure spaces and investigate the possible directions at which we can see recurrence. Extending the notion of recurrence to the unit circle, we consider the structure of the set of recurrent directions. This is joint work with Dan Rudolph and Ayse Sahin.

Uijin Jung , Korea Advanced Institute of Science and Technology, Daejeon
Extension theorems for codes between shifts of finite type

Byungsoo Kim ,   Colorado State University
Rolling molecules: stationray states, stability and statistical physics
When molecules of a gas or liquid move along a surface, it is commonly assumed that the molecular motion is purely translational along that surface. On the other hand, recent studies of molecular mono-layers and molecular devices indicate a strong possibility of rolling motion as well. In this talk, we consider the limiting case in which the rolling motion dominates the translational motion. The molecular monolayer is represented as an interacting system of rolling balls, with off-center center of mass and interacting through the charges and dipoles that are positioned in the center of mass. Two ranges of parameters for the system are studied: pure Lennard-Jones interaction that prevents particles from colliding, and dipole+Lennard-Jones, modeling water monolayers. We consider stationary states of the system and their stability. Next, we analyze (numerically) long-term motion of an ensemble of these molecules and show that classical notions like temperature can be applied to a system with even a small number of balls.

Patrick LaVictoire ,   U.C.Berkeley
PhD student, advised by Michael Christ
Singular Integral Theory and L^1 Ergodic Theorems
The harmonic analysis approach to proving nonstandard ergodic theorems has been remarkably successful. By transferring maximal and oscillation inequalities from dynamical systems to the integers, one finds many such problems transformed into questions of singular integral estimates. In this talk, I will sketch several instances of this approach, focusing on a technique of Christ and Fefferman via which several nonstandard ergodic theorems of interest have recently been extended from L^2 to L^1.

Douglas Lind ,   University of Washington and Yale University
Specification for Algebraic Actions
Specification is an important orbit tracing property of group actions. For expansive actions by automorphisms of compact abelian groups I'll describe the role homoclinic points play in establishing specification, and discuss recent joint work with Schmidt and Verbitskiy extending these ideas to certain nonexpansive higher rank actions.

Elon Lindenstrauss ,   Hebrew University and Princeton University
Classifying invariant measures on homogeneous spaces
A well known result of Rudolph states that Lebesgue measure is the unique x2, x3 invariant and ergodic measure on R/Z with positive entropy. Extending this theorem to diagonalizable flows on homogeneous spaces turned out to require rather different tools, and I will explain one such tool- the low entropy method - and how it relates to another important paper of Rudolph on the ergodic theoretic properties of the Patterson Sullivan measure on geometrically finite hyperbolic surfaces.

Alejandro Maass ,   University of Chile, Santiago
Nilsequences and a structure theorem for topological dynamical systems
In this talk, a characterization of inverse limits of nilsystems in topological dynamics via a structure theorem for topological dynamical systems is given. This is an analog of the structure theorem for measure preserving systems constructed by B. Host and B. Kra. Two applications of the structure are discussed. The firrst is to nilsequences: we give a characterization that detects if a given sequence is a nilsequence by only testing properties on finite intervals. The second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system. We show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1. (This is a joint work with Bernard Host and Bryna Kra.)

Don Ornstein ,   Stanford University
(Retrospective on some of Dan Rudolph's work)
The speaker will discuss some of Dan Rudolph's early work including his work on the Bernoulli isomorphism theory and some of his ingenious counter exambles.

Ayse Sahin ,   DePaul University
The geometry and directional entropy of rank one Z^d actions
We study the dynamical properties of rank one Z^d actions as a function of the geometry of the shapes of the towers generating the action. The main results relate the geometry to directional entropy and the orbit equivalence classification of the action. This is joint work with E. A. Robinson.

Klaus Schmidt ,   University of Vienna and Erwin Schrodinger Institute
Principal Algebraic Actions
Let G be a countable discrete group. Every element f in the integral group ring ZG of G acts by right convolution rho_f on the compact group T^G of all maps from G to T=R/Z. The corresponding 'principal algebraic action' alpha_f of G is the (left) shift action of G on the subgroup X_f of T^G which is the kernel of rho_f. This talk is about elementary dynamical properties of this class of actions, and is largely based on joint work with Christopher Deninger and/or Doug Lind.

Bethany Springer ,   Colorado State University
PhD student, advised by Daniel Rudolph
New results in nearly continuous Kakutani equivalence
It was previously known that a conjugacy between induced systems on Polish probability spaces (X,T,\mu) and (Y,S, \nu) on nearly clopen subsets $A \subset X$ and $B \subset Y$ of same relative measure can be extended to a measurable orbit equivalence between the systems. This talk gives an algorithm for modifying the conjugacy in order to establish nearly continuous Kakutani equivalence. We also show that in the case $\mu(A) > \nu(B)$, we can establish conjugacy between $T$ and an induced system $S_{\bar B}$ where the systems are nearly uniquely ergodic.

Jean-Paul Thouvenot ,   Universite' de Paris VI, France
A criterion for a transformation to satisfy the weak Pinsker Property
The proofs of the deep structure theorems of Dan Rudolph concerning the finite (and isometric) extensions of Bernoulli shifts relied on the identification of \bar d separated distributions which had then to be extremal. We shall describe a criterion for a process which allows to produce separated distributions in a more general situation than the compact case, equivalent to the weak Pinsker property.

Raul Ures ,   University of the Republic, Montevideo, Uruguay
Entropy maximizing measures for some partially hyperbolic systems