Mahsa Allahbakhshi, University of Victoria
PhD student, advised by Anthony Quas
Factor triples and relative entropy
For an abstract, click here.
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
For an abstract, click here.
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
For the abstract, click
here.
Byungsoo Kim ,
Colorado State University
PhD student, advised by Vakhtang Putkaradze
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
For the abstract, click here .
Ben Webb , Georgia Tech
PhD student, advised by Leonid Bunimovich
Isospectral Graph Reduction and Estimates of Matrices' Spectra.
Motivated by recent results in the area of dynamical networks,
we consider a general procedure for reducing the size and structural
complexity of a graph while maintaining its spectrum up to some set
known a priori. The proposed procedure has allot of flexibility and can
be used to establish new relations between the structure of graphs.
Furthermore, this procedure improves the classic eigenvalue estimates of
Gershgorin, Brauer, and Brualdi for square matrices with complex valued
entries.
Benjamin Weiss , Hebrew University
On Dan Rudolph's impact on measurable dynamics
This will be a personal account of some of Dan's
contributions to measurable dynamics. Since I am sure that
a great deal will have already been described by the
other speakers at this meeting I will concentrate mainly
on the complement - which is ill defined at the present.