Schedule
The dynamics seminar meets from 3-4 in room 206.
Fall 2011
January 30: Amie Wilkinson (Chicago)
Title: Absolute continuity,
exponents and rigidity
Abstract: The geodesics in a compact surface of negative
curvature
display stability properties originating in the chaotic, hyperbolic
nature of the geodesic flow on the associated unit tangent bundle.
Considered as a foliation of this bundle, this collection of geodesics
persists in a strong way when one perturbs of the Riemannian metric,
or the geodesic flow generated by this metric, or even the time-one
map of this flow: for any perturbed system there is a corresponding
"shadow foliation" with one-dimensional smooth leaves that is
homeomorphic to the original geodesic foliation. A counterpart to
this foliation stability is a curious rigidity phenomenon that arises
when one studies the disintegration of volume along the leaves of this
perturbed shadow foliation. I will describe this phenomenon and its
underlying causes. This is recent work with Artur Avila and Marcelo
Viana.
February 6: Van Cyr (Northwestern)
Title: A Z^2 generalization of the Morse-Hedlund Theorem.
Abstract: Attached
February 20: Vadim Kaloshin (Maryland)
Title: Arnold diffusion via Normally Hyperbolic Invariant
Cylinders.
Abstract:
In 1964 Arnold constructed an example of instabilities for nearly
integrable
systems and conjectured that generically this phenomenon takes place.
There has been some progress attacking this conjecture in the past
decade.
Jointly with Ke Zhang we present a new approach to solve this problem.
It is based on a construction of crumpled and flower Normally Hyperbolic
Invariant Cylinders with kissing property. Then to construct diffusion
along
these cylinders we apply Mather variational mechanism. A part of the
project
is also joint with P. Bernard.
February 27: Natalie McGathey (Purdue North Central)
Title: Invariant Measures and Homeomorphisms of Boundaries
Abstract: An important question in ergodic theory is: given an
action of a
group $G$ on a space $X$, classify all (ergodic) invariant probability
measures for this action. We will give such a classification for the
case
$G = \mathrm{PSL}_2(\mathbb{R}), \mathrm{Isom}_+\mathbb{H}_K^n, where
$K$
denotes $\mathbb{R},\ \mathbb{C},\ \mathbb{H},$ or $\mathbb{O}$ with
$n=2$, and $X$ is $L/G$ or some large group $L$. In this talk, we will
briefly describe how this particular setting is motivated by trying to
understand embeddings of Teichmuller space; we will then state the
classification results and give some flavor of the proofs.
March 5: Patrick LaVictoire (Wisconsin)
Title: Pointwise ergodic theorems on virtually nilpotent groups
Abstract: Starting with the work of Bourgain, a "quantitative"
harmonic
analysis approach has proved many nonstandard pointwise ergodic
theorems,
recently including several results for sparse averages of L^1 functions.
While the original methods worked via Fourier analysis, some of them can
be
reframed in terms of additive combinatorics, and thereby extended to
actions of virtually nilpotent discrete groups. This is joint work with
J.
Rosenblatt and A. Parrish.
March 12: Lin Shu (Peking)
Title: Folding entropy and dimension theory for invariant
measures
of endomorphisms
Abstract: Consider a $C^2$ non-invertible but non-degenerate
endomorphism $f$
on a compact Riemannian manifold without boundary. We are interested in
the
dimension theories of $f$- invariant measures. By using the notion of
folding
entropy introduced by Ruelle, we set up an equality relating entropy,
folding
entropy and negative Lyapunov exponents. Based on this, we establish
the exact
dimensional property of an ergodic hyperbolic measure. We also give a
new formula
of Lyapunov
dimension of ergodic measures and show it coincides with the dimension
of hyperbolic
ergodic measures in a setting of random endomorphisms. An application
of folding
entropy to the theory of entropy production is also addressed.
Our results extend several well known results of Ledrappier-Young and
Barreira et
al. for diffeomorphisms to the case of
endomorphisms.
April 16: Jim Tseng (UIUC)
Title: Dense and nondense orbits for pairs of maps
Abstract: For a pair of hyperbolic commuting linear maps of the
torus, we show that the set of points dense under one map and nondense
under the other is of full Hausdorff dimension. In dimension two, we
can also show weaker results for noncommuting maps. Our technique for
commuting maps can be applied to other spaces. This is joint work
with V. Bergelson and M. Einsiedler.
April 30: Michele Lee (Michigan/Maryland)
Title: Dynamics on the PSL(2, C)-character variety of certain
hyperbolic
3-manifolds
Abstract: The PSL(2, C)-character variety of a hyperbolic
3-manifold M is
essentially the set of homomorphisms of the fundamental group of M into
PSL(2, C), up to conjugacy. We will discuss the action of the group of
outer automorphisms of the fundamental group of M on this space. In
particular, we will discuss how one can find domains of discontinuity
for the action.
May 21: Keith Burns (Northwestern)
Title: TBA