Abstracts, Trends in Dynamics
Northwestern University
April 27May 1, 2011

Tim Austin, Brown University
Title: Multiple recurrence and joining rigidity
Since Furstenberg proved the Multiple Recurrence Theorem
and its equivalence to Szemeredi's Theorem, ergodic theory has seen
the rise of a large subarea dedicated to the study of multiple
recurrence phenomena and the structures within probabilitypreserving
systems that give rise to them. In this talk I will discuss some
recent advances in the understanding of convergence of certain
`nonconventional' ergodic averages that play a central role in this
theory, and in particular will relate their analysis to a search for
rigidity among those joinings of general probabilitypreserving
systems that exhibit some additional symmetries beyond the diagonal
transformations coming from the original system.

Uri Bader, Technion University
Title: Weyl groups, FurstenbergPoisson boundaries and linear representations
I'll explain how to attach to a given group L a group W,
such that for each linear representation of L we get a map from W to the Weyl group of the Zariski closure of the image.
Our method involves the FurstenbergPoisson boundary of L and a new ergodic theorem.
If time permits, I'll explain how to derive some classical theorems, e.g Margulis' superigidity, from this result.
This is a joint work with Alex Furman.

Yves Benoist, Orsay University
Title: Random walks on homogeneous spaces
I will present a joint work with JeanFrancois Quint, extending previous results
by A. Eskin and G. Margulis, and answering their conjectures:
A random walk on a finite volume homogeneous space is always recurrent
as soon as the transition probability has finite exponential moments and
its support generates a subgroup whose Zariski closure is semisimple.

Christian Bonatti , University of Bourgogne
Title: Toward a global view of dynamical systems for the C^1 topology
Recent results (from the last 15 years) lead to the feeling
that one could divide the space of diffeomorphisms
by using robust local phenomena (as the famous transverse homoclinic
intersections discovered by Poincare and characterizing (that is a
recent result) the chaotic dynamics) and global uniform and robust
structures (as the hyperbolicity introduced by AnosovSmale, partial
hyperbolicity , filtrations, etc..)
Structures avoid some local phenomenon: this devides the space of
diffeomorphisms in disjoint open sets, whose union is (conjecturally)
dense.

Lewis Bowen , Texas A & M University
Title: Entropy for sofic groups
In 1958, Kolmogorov defined the entropy of a probability measure
preserving transformation. In the 70s and 80s
researchers extended entropy theory to measure preserving actions of
amenable groups (Kieffer, OrnsteinWeiss). My recent work generalizes the
entropy concept to actions of sofic groups; a class of groups that
contains for example, all subgroups of GL(n,C). Applications include the
classification of Bernoulli shifts over a free group, answering a
question of Ornstein and Weiss.

Danny Calegari , Caltech University
Title: Random rigidity in the free group
We prove a rigidity theorem for the geometry of the unit ball in the stable commutator length norm spanned by k random elements of the commutator subgroup of a free group of fixed big length n; such unit balls are C^0 close to regular octahedra. A heuristic argument suggests that the same is true in all hyperbolic groups. This is joint work with Alden Walker.

Serge Cantat , University of Rennes
Title: The Cremona group
The Cremona group is the group of
all birational transformations of the plane.
Its dimension is infinite, and it contains elements
with interesting dynamics, for example Henon mappings.
I shall describe recent results that describe the algebraic
properties of this group. The proofs make use of basic
algebraic geometry, dynamical systems, and geometric group
theory.

Matthew Foreman , University of California, Irvine
Title: Classifying measure preserving diffeomorphisms of the torus: Symbolic systems and the AnosovKatok method
Relatively recent joint work with Rudolph and Weiss showed rigorously that it is impossible to classify ergodic measure preserving transformations of the unit interval. This talk will show how to extend these impossibility results to measure preserving diffeomorphisms of the torus using the AnosovKatok technique. The word "impossible" will be defined and there will be a discussion of "circular systems"those symbolic systems that arise from the AnosovKatok technique. The new results are joint with B. Weiss.

Giovanni Forni , University of Maryland
Title: Limit Distributions for Horocycle Flows
We prove limit theorems for horocycle flows on compact surfaces
of constant negative curvature. The argument is based on the construction
of a special family of horocycleinvariant finitely additive Hoelder measures on rectifiable arcs. An asymptotic formula for time integrals for
horocycle flows is obtained in terms of the finitelyadditive measures, and limit theorems follow as a corollary of the asymptotic formula.
This is joint work with A. Bufetov.

Nikos Frantzikinakis , University of Crete
Title: Some new multiple ergodic theorems and related open problems
In recent years there has been a lot of interest in studying the limiting behavior
of multiple ergodic averages involving iterates given by a variety of sequences: polynomial,
random, [smooth], sequences involving the primes etc. Despite the various successes and
the various new tools that have been developed, several natural problems remain unsolved.
I am going to briefly mention these new tools, some recent results, and a few open problems.

Ursula Hamenstadt , University of Bonn
Title: Dynamics of the Teichmueller flow on strata of quadratic differentials
We use symbolic dynamics to investigate dynamical properties
of the Teichmueller flow in strata. This implies a sharp counting resut for
periodic orbits in the thin part of strata.

Bernard Host , University of Paris East
Title: Some open questions about higher order correlations
In ergodic theory, correlation sequences are well understood, but
little is known about higher order correlations. In this talk,
we give an overview of some questions about these sequences,
with few answers (and most of them negative). This is joint work with
Bryna Kra.

Vadim Kaloshin , University of Maryland
Title: On Herman's oldest open question in dynamics and diffusion along mean motion resonance for the restricted planar three body problem (joint with J. Fejoz, M. Giardia, P. Roldan)
Herman's oldest open question in dynamics, in particular, claims that for an open dense set of initial conditions of a 3 body problem orbits are unbounded. We make a small step toward this conjecture. We consider a planar 3 body problem so that one body is large (the Sun), one small (Jupiter), and the other is tiny (Asteroid). In the regime when period of Jupiter and of Asteroid are in resonance (called mean motion resonance) 1:7 we show that orbit of Asteroid can substantially change its shape. This 1:7 resonance is close to mean motion resonance between Jupiter and Uranus. The proof is computer assisted with fundamental role played by apriori unstable structure of the underlying Hamiltonian system.

Anatole Katok (CANCELLED) , Penn State University
Title: KAM and rigidity
Classical KAM method that was originally developed for finding elements of elliptic behavior in Hamiltonian systems, has found surprising applications in both partially hyperbolic and parabolic dynamics. In this talk I will give an overview of results from joint works with (separately) Danijela Damjanovic and Zhenqi Jenny Wang. Local differentiable rigidity problem for algebraic partially hyperbolic actions of higher rank abelian groups has been solved more or less completely, and in the parabolic situation rigidity for finitely parametric families of perturbations has been established in a variety or representative cases.

Alejandro Maass , University of Chile
Title: Regionally proximal relations, independence and nilsystems
The notion of dynamical parallelepiped was introduced in
topological dynamics inspired in the proofs of nonconventional ergodic
averages by
Host and Kra and its extensions. Among other properties, these
parallelepipeds allow to define regionally proximal relations of higher
order which characterize the maximal nilfactors of a topological dynamical
system. In this talk we will look to the regionally proximal relations from
the local complexity theory point of view. In particular, we will see that
the "nil part" of a topological dynamical system is characterized by special
types of recurrence with respect to open covers.

Andres Navas , University of Santiago
Title: On the (bary)center technique for the study of cocycles of isometries
We consider cocycles of isometries of a nonpositively curved
space over a minimal dynamics. We show that the classical (bary)center
technique yields a generalized version of the GottschalkHedlund theorem:
the existence of a bounded orbit is equivalent to that of a continuous invariant
section. We also discuss several interesting examples for which no bounded
orbit exists (those systems cannot be minimal, though they may be
topologically transitive). Finally, we apply these ideas to deal with
uniformly nonhyperbolic cocycles, thus showing that they are
close to "reducible" ones.

Enrique Pujals (CANCELLED) , IMPA
Title: Some simple questions and results related to the C^r
stability conjecture
We will expose some questions related to the C^r
structural stability conjecture for surface diffeomorphisms, and we
will try to explain some partial results.

Federico Rodriguez Hertz , IMERL and Penn State University
Title: Global Rigidity for certain actions of higher rank lattices on
the torus
In this talk we will give an approach to the following
theorem:
Let $\Gamma$ be an irreducible lattice in a connected semisimple Lie
group with finite center, no nontrivial compact factor and of rank
bigger than one. Let $a:\Gamma \to Diff(T^N)$ be a real analytic
action on the torus preserving an ergodic large measure (large means
essentially that its support is non trivial in homotopy). $a$
induces a representation $a_0:\Gamma \to SL(N,Z)$. Assume further that
$a_0$ has no zero weight and no rank one factor. Then $a$ and $a_0$
are conjugated by a real analytic map outside a finite $a_0$ invariant
set.
The theorem essentially says that nonlinear action $a$ is
built from linear $a_0$ by blowing
up finitely many point.
This is joint work with A. Gorodnik, B. Kalinin and A. Katok.

Omri Sarig , Weizmann Institute and Penn State University
Title: Countable Markov partitions for smooth surface diffeomorphisms with positive topological entropy
Abstract: Suppose f:M>M is a C^{1+epsilon} surface diffeomorphism with topological entropy h>0.
For every positive delta less than h,
we construct a "delta large" invariant set E such that fE admits a countable Markov partition.
"Delta large" means that E has full measure for every ergodic invariant measure with entropy bigger than delta.

Joseph Silverman , Brown University
Title: Dynamical moduli spaces and the locus of critically finite maps
The moduli space M_d of rational maps f(z) of degree d modulo the
conjugation action of PGL_2 plays an important role in the study of
the dynamics of such maps. In this talk I will discuss known results
and describe some open problems regarding these moduli spaces,
including the construction of M_d and its completion via geometric
invariant theory, the isomorphism M_2 = A^2 and the rationality of M_d
for higher d, and the geometry and arithmetic of the special set of
points in M_d corresponding to maps f(z) having finite critical
orbits.

LaiSang Young , New York University
Title: Toward a smooth ergodic theory for infinite dimensional systems
I will discuss some first steps toward building a nonuniform hyperbolic theory
for infinite dimensional dynamical systems, focusing on settings that are
consistent with those in systems defined by dissipative parabolic PDEs.

Mike Zieve , University of Michigan
Title: From dynamics to number theory and back again
In recent years, new connections between complex dynamics
and number theory have yielded advances in both subjects. I will
present several instances of this. In particular, I will determine the
pairs of complex polynomials having forward orbits with infinite
intersection, I will show that any polynomial with rational
coefficients induces a map on the rational numbers which is at most
4to1 outside a finite set, and I will explain how these results suggest
conjectural dynamical generalizations of some fundamental results in
number theory (namely, the Mordell conjecture and Mazur's theorem
on rational torsion on elliptic curves).