Abstracts, Midwest Dynamical Systems
October 29-31, 2010
Ciprian Demeter,   Indiana University, Bloomington
Proof of the HRT conjecture for special configurations
The strong HRT conjecture asserts that the time-frequency
translates of any nontrivial function in L^2(R) are linearly
independent. The weak HRT conjecture has the same formulation, but this
time for Schwartz functions. Prior to our
work, the only result on HRT of a reasonably general nature was Linnell's
proof in the case when the translates belong to a
lattice. I will briefly describe an alternative argument to Linnell's
(joint work with Zubin Gautam), inspired by the theory
of random Schrodinger operators. Then I will explore a number
theoretical approach to the HRT conjecture, for some special 4 point
Van Cyr,   Northwestern University
Constructing Conformal Measures for Transient Markov Shifts
In this talk we will explore the thermodynamic formalism of countable Markov
shifts exhibiting transience. The main goal will be to construct a conformal mea-
sure for any locally compact transient shift using insight from the theory of Martin
boundaries for transient Markov chains. If time permits, some recent results con-
cerning the general (not locally compact) case will be discussed.
David Fisher,   Indiana University
Local rigidity revisited
After giving some background, history and motivation,
I will explain a new, simplified proof of some results on local
rigidity of group actions. This result applies to a large class of
including all affine algebraic actions of cocompact lattices in
noncompact simple Lie groups with property (T). The proof is
simple, but does require the use of ultrafilters. I will attempt
make ultrafilters user friendly.
John Franks,   Northwestern University
Area Preserving Entropy Zero Diffeomorphisms
This talk will discuss joint work with Michael Handel
concerning the dynamic structure of area preserving
diffeomorphisms of the two-sphere which have zero
topological entropy. The intended applications are
to show the existence of global fixed points for certain
subgroups of the centralizer of such a diffeomorphism.
These fixed points in turn can be used to show that
the actions of groups like the mapping class group
are very restricted.
Alex Furman,   University of Illinois, Chicago
The space of metrics on a hyperbolic group,
Sullivan's measure, and rough symmetries.
Many concepts associated to the geometry and dynamics
of a negatively curved closed manifold, such as, topological
entropy, lengths of closed geodesics, Sullivan's measure on
the space of geodesics, admit a generalization to the framework
of proper left invariant metrics on a Gromov hyperbolic group,
taken up to bounded distance.
This broader framework includes such objects as word metrics,
and distances coming from convex cocompact actions of CAT(0)
We shall overview this broader framework and show how one
might single out fundamental groups of locally symmetric spaces
and the locally symmetric metrics, among all other examples.
Patrick LaVictoire,   University of Wisconsin
Exponential Sums and Pointwise Convergence of Averages
The convergence, in several senses, of a sequence of weighted
ergodic averages depends greatly on the behavior of an associated sequence
of exponential sums. In the past few years, new results on the pointwise
convergence of weighted averages for L^1 functions have highlighted this
principle, especially when passage to a subsequence is permitted. I will
illustrate two of these results, one positive and one negative, and a
threshold between them in the averages along a perturbation of the square
Charles Pugh,   Northwestern University
This is joint work with Conan Wu. It concerns funnels of solutions to
continuous ODE's. The ODE's have non-unique solutions and one wants
information about the topology of the cross section of the solution funnel.
(The funnel consists of all solutions to the ODE with a given initial
condition. The cross section consists of the points attained by these
solutions at a fixed time T.) We have some new results and new questions.
Ana Rechtman,   Northwestern University
Existence of periodic orbits for geodesible flows on 3-manifolds
Geodesible flows are those for which it is possible to find a Riemannian
metric on the ambient manifold that makes all the orbits geodesics. We
will study geodesible flows on closed 3-manifolds, in particular, the
existence of periodic orbits: by adding the hypothesis of the existence of
an invariant volume, the flow possesses a periodic orbit if the ambient
manifold is not a torus bundle over the circle. Moreover, if the manifold
is the 3-sphere there is an unknotted periodic orbit.
Victoria Sadovskaya,   University of Southern Alabama
Linear cocycles over hyperbolic systems:
pinching, conformality, and cohomology
We consider a linear cocycle over an Anosov diffeomorphism $f$,
i.e. an automorphism of a vector bundle that projects to $f$.
An important example is given by the differential $Df$ or its
restriction to an invariant sub-bundle. For Holder continuous
cocycles satisfying pinching or periodic data assumptions,
we establish existence and continuity of measurable invariant
sub-bundles and conformal structures. We obtain criteria for
a cocycle to be isometric or conformal in terms of its periodic
data and discuss applications of such results.