on Dynamical Systems and Related Topics

University of Maryland, March 20-23, 2004

**
Scot Adams,
**
University of Minnesota
*
From Lorentzian dynamics to the decay of matrix coefficients
*

The Howe-Moore theorem states that any ergodic action of a connected
noncompact, finite-center, simple Lie group is mixing. Thus, for
example, ergodicity inherits to all noncompact closed subgroups, a
very useful fact which immediately yields ergodicity of many
geometrically-motivated actions. The proofs I know of Howe-Moore go
via unitary representation theory. By thinking of a Hilbert space as
an infinite-dimensional Riemannian manifold and by adjusting
techniques originally used in studying Lorentzian (and Riemannian)
dynamics, we can obtain a version of Howe-Moore that is valid for all
connected Lie groups, though only useful for nonAbelian
groups. Specifically, for any connected Lie group $G$, for any faithful
irreducible unitary representation of $G$, we have: any matrix
coefficient tends to zero, as $Ad(g)$ leaves compact subsets of
$GL({\frak g})$, where ${\frak g}$ is the Lie algebra of $G$.

To view slides of this talk, click
here .

For a short note closely related to the talk,
click
here.

**
Karen Ball
**,
Institute for Mathematics and its Applications, and Indiana University
*
Monotone factors of Bernoulli shifts
*

Let B(p) and B(q) be Bernoulli shifts on {0,1}^Z, with p and q the
probability of a 1 in each system. If h(p) > h(q),
we know that there is a factor map taking B(p)
to B(q). If in addition, p > q, we can ask whether there is a
factor map f which is monotone: f(x)_i <= x_i for each coordinate i
of almost every point x. In this talk, we show the stronger
result that there is a monotone finitary code from B(p) to B(q).

**
Maria Alice Bertolim
**,
Sao Paolo State University at Campinas (UNICAMP), Brazil
*
Dynamical and Topological Aspects of Lyapunov Graphs and
the Poincare-Hopf Inequalities
*

We present the interplay between topological dynamical
systems theory with network flow theory in order to obtain a
continuation result for abstract Lyapunov graphs. The
Poincare-Hopf inequalities are shown to be necessary and
sufficient conditions for this continuation. We relate these
inequalities to the Morse inequalities. The Morse polytope
determined by the Morse inequalities is presented together with
some of its geometrical properties.

**
Sergey Bezuglyi**,
University of New South Wales and Institute for Low Temperature
Physics, Kharkov, Ukraine
*
Approximation of homeomorphisms of a Cantor set
*

On the group of all homeomorphisms of a Cantor set two topologies, which
have their origins in ergodic theory, are defined and studied.
Various classes of homeomorphisms (for example, odometers, minimal,
aperiodic) are considered and the closures of these classes are found.
The concept of Bratteli diagrams is discussed for aperiodic
homeomorphisms.

**
Kristian Bjerklov **,
Royal Institute of Technology, Sweden
*
Quasi-periodic Schroedinger equations
*

In this talk I will present recent results concerning
the quasi-periodic Schroedinger equation; both the
time-continuous and its discrete analog.
Under mild regularity assumptions on
the potential function I obtain (in
the large coupling regime) lower estimates
on the Lyapunov exponent and the measure of the spectrum of the associated
Schroedinger operator. Moreover, these equations
also induce diffeomorphisms of
the two-torus. I show that they can be minimal and non-ergodic.

**
Alexander Blokh**,
University of Alabama at Birmingham
*
Wandering triangles exist
*

In his preprint "On the combinatorics and dynamics of
iterated rational maps" (Princeton, 1985),
W. P. Thurston developed the
theory of invariant laminations. He studied connections between the
dynamics of complex polynomials on their Julia sets and the dynamics
of the map $\sigma_d(z)=z^d$ on the unit circle. The following notion
was introduced in the preprint: a set $A_0=\{x_0,y_0,z_0\}$ on the
unit circle is called a wandering triangle (for $\sigma_d$)
if the following holds:

(1) each set $\sigma_d^n(A_0), n=0, 1, \dots$ consists of three distinct
points, and

(2) the convex hulls of all sets $\sigma_d^n(A_0), n=0, 1, \dots$ in
the unit disk are pairwise disjoint.

Thurston proved in his preprint that $\sigma_2$ admits no
wandering triangles and asked if they exist for $\sigma_d, d>2$. We
obtain the following Theorem: for any $d>2$ there exist uncountably many
$\sigma_d$-invariant laminations with wandering triangles.

**
Angela Desai**,
University of Maryland

(PhD student, advised by
Mike Boyle)
*
Z^d shifts of finite type and sofic systems:
epsilon-control of some entropies
*

In certain coarse respects, Z^d SFT's and sofic shifts
must behave like their Z counterparts.

(1) Let S be a Z^d (d>1) SFT. Then the entropies
of SFT subshifts of S are dense in [0,h(S)].
This extends the result of Quas and Trow that
h(S) is an accumulation point of such entropies.

(2) Let S be a Z^d (d>1) sofic shift. Then the entropies
of sofic subshifts of S are dense in [0,h(S)].

(3) If S is a Z^d sofic shift and epsilon > 0,
then there is a Z^d SFT T which covers S and
satisfies h(T) < h(S) + epsilon. This is answers
a question of Benjy Weiss. (We can't answer his
question, can this epsilon be made 0.)

**
Manfred Einsiedler
**,
University of Washington
*
Measure rigidity, disjointness, and
rigidity of factors for commuting torus automorphisms
*

Furstenberg's open problem about $\times 2,\times 3$-invariant
measures on the circle group has motivated much interesting work.
Rudolph's partial result classified the Lebesgue measure to be the
only invariant and ergodic probability measure with positive entropy.
A. Katok and Spatzier and later Kalinin and A. Katok extended this
result to certain actions on higher dimensional tori (e.g. two
commuting automorphisms of the three-dimensional torus) but in
general an additional (ergodicity-type) assumption was needed.

In joint work with E.Lindenstrauss we avoid additional assumptions
and show for an invariant and ergodic probability measure with
respect to a higher rank action on a torus (solenoid) that positive
entropy implies that the measure is (at least partially) algebraic.

This can be applied to completely classify factors and disjointness
of such actions (with respect to their Haar measures).

To view slides of this talk, click
here .

**
David Fisher **,
Lehman College -- CUNY
*
Local Rigidity and KAM theory
*

I will discuss recent joint work with G.A.Margulis, in which we have
dramatically improved regularity of the
conjugacy in some of our results concerning local rigidity of group
actions. Though this work is part of a
series of results concerning rigidity of quasi-affine actions of higher
rank lattices, I will only discuss the following
theorem, in order to be able to describe some connnections to KAM theory:

Theorem: Let G be a group with property T of Kazhdan and M a
compact manifold. Let A be an isometric
action of G on M. Then any other C^{\infty} action of G on M
which is sufficiently close to A in the
C^{\infty} topology is conjugate to A by a C^{\infty}
diffeomorphism f.

Though the proof borrows ideas from KAM theory, one cannot in fact prove
the general theorem by the KAM method. I will explain why this is true
when A is the trivial action. If time permits, I will describe some
work in progress, which is an attempt
to prove local rigidity theorems for "smaller" groups using a KAM method.

**
Todd Fisher **,
Northwestern University

(PhD student, advised by
Amie Wilkinson )
*
The Structure of Hyperbolic Sets
*

This talk will address various aspects on the structure
of hyperbolic set. First, I will discuss hyperbolic sets that are not
included in any locally maximal hyperbolic sets. Specifically,
the
existence of a Markov partition for hyperbolic sets that are not locally
maximal will be
examined.
Second, it will be shown that a hyperbolic set with interior plus
some additional assumptions is Anosov.
Lastly, examples will be given of hyperbolic sets
with nonempty interior that are not Anosov.

**
Travis Fisher**,
Penn State University

(PhD student, advised by
Anatole Katok)
*
Differentiable Rigidity for Non-Semisimple Toral Actions
*

We consider an action by Z^d by hyperbolic toral
automorphisms which has no rank one factors. Assuming no Lyapunov
exponents are rationally proportional, any C^1 perturbation of this action
is C^infinity conjugate to the original action. This extends a result of
Katok and Spatzier which required the action to be semisimple. I will
discuss the new methods needed to handle non-trivial Jordan blocks. This
is joint work with Manfred Einsiedler.

**
Nikos Frantzikinakis**,
Penn State University
*
Polynomial Ergodic Averages Converge to the Product of the Integrals
*

In 1996 Furstenberg and Weiss proved that for a totally ergodic
system the average of a product of functions evaluated along the times n
and n^2 converges in the mean to the product of the integrals. Bergelson
asked whether the same is true for any finite number of polynomials with
pairwise differing degrees. We prove that this is true and we deduce a
strong multiple recurrence property for totally ergodic systems. This is
joint work with B. Kra.

**
Anish Ghosh**,
Brandeis University

(PhD student, advised by
Dmitry Kleinbock)
*
Dynamics on SL(n,R)/SL(n,Z) and
Diophantine Approximation on Affine Subspaces
*

For the abstract, click
here.

**
Arek Goetz ,
**
San Francisco State University
*
Natural coexistence of transitive and periodic components in a
planar piecewise rotation
*

In a joint work with Peter Ashwin (GB) we constructed a map
T that is a permutation of two dimensional cones.
The map T fixes an irregular pentagon consisting of an infinite number
of periodic cells as well it fixes its complement - a continuum of
invariant polygonal curves on which the action of T is a uniquely
ergodic interval exchange transformation. The parameters defining T
lie in the cyclotomic field obtained by adjoining to the rational
numbers a fifth root of
unity.

**
Chris Hoffman
**,
University of Washington
*
Phase transitions in one dimensional systems
*

This is joint work with Noam Berger and Vladas Sidoravicius.

Let f be an expanding map of the unit circle which is in
$C^{1+\epsilon}$. It is well known that f has a unique invariant
absolutely continuous measure and the natural extension of this system is
isomorphic to a Bernoulli shift. However Quas has shown that if f is
only assumed to be in $C^1$ then f may have multiple absolutely
continuous invariant measures with a variety of ergodic theoretic
properties.

I will talk about a class of probabilistic models that are generalizations
of mixing Markov chains. These are models generated by transition
functions which are regular and continuous. These models exhibit similar
behavior to expanding maps of the circle. That is, if the transition
function has a certain modulus of continuity, then there is a unique
stationary measure which is consistent with the transition function.
This measure also has "nice" ergodic theoretic properties. But if it does
not then it is possible that there are multiple invariant measures
which are consistent with the transition function. These measures may
have many possible ergodic theoretic properties.

**
Danrun Huang ,
**
St. Cloud State University
*
Some remarks on Restorff's Classification of Cuntz-Krieger algebras
*

Gunnar Restorff (University of Copenhagen) has recently classified all
nonsimple Cuntz-Krieger algebras up to stable isomorphism, using the
techniques from both symbolic dynamics and C*-algebras.
In this talk, I will try to find solutions to some of the open questions
in his paper.

**
Boris Kalinin,
**
University of South Alabama
*
On smooth classification of Cartan actions of R^k and Z^k
*

Anosov actions of Z^k and R^k arising from natural algebraic
constructions have been extensively studied recently. In contrast
to Anosov diffeomorphisms and flows, the higher rank actions exhibit
such remarkable properties as rigidity of invariant measures and
rigidity
of measure preserving isomorphisms. These algebraic actions are often
locally rigid, i.e. smoothly conjugate to any small perturbation. In
this
talk I will discuss the problem of smooth classification of nonalgebraic
actions, i.e. the existence of a smooth conjugacy to an algebraic model.
The main result is such a classification for certain classes of Cartan
actions obtained in my joint work with R. Spatzier.

**
Mike Keane
**,
Wesleyan University

mkeane@wesleyan.edu
*
Some new finitary codes
*

This is joint work with Professor T. Hamachi of Kyushu University. We
explain a new method to construct finitary codes and discuss an
example and some work in progress concerning this method.

**
Bryna Kra**,
Penn State University
*
Multiple ergodic averages
*

I will give an overview of recent results on nonconventional
ergodic averages, including ones along arithmetic progressions,
polynomials and a generalization for commuting transformations. I will
discuss the relation of these results and combinatorial applications.

To view slides of this talk, click
here .

**
Alejandro Maass**,
Universidad de Chile
*
Eigenvalues of Linearly Recurrent Cantor Dynamical Systems and
Generalisations to some Tiling Systems
*

We give necessary and sufficient conditions to have measurable and
continuous eigenfunctions for linearly recurrent
Cantor dynamical systems.
We also construct explicitely an example of linearly recurrent system
with a non-trivial Kronecker factor and a trivial equicontinuous factor.
Finally we give some extensions to minimal free $Z^d$-actions on the
Cantor set.

**
Alica Miller
**,
University of Illinois at Urbana-Champaign
*
Orbit - equivalence of compact minimal R-flows
*

In this talk we will discuss some questions about orbit -
equivalence of compact minimal R-flows, like, for example: a)
classification of regularly almost periodic flows up to orbit -
equivalence; b) orbit - equivalence of almost periodic and regularly
almost periodic flows with the weakly mixing ones and the cocycles by
which this orbit - equivalence can be achieved; c) flows orbit -
equivalent with distal ones.
(Joint work in progress with Joe Rosenblatt.)

**
Sheldon Newhouse
**,
Michigan State University
*
Symbolic Extensions and smoothness
*

Recently, there have been some powerful methods developed by
Boyle, Downarowicz, D. Fiebig, U. Fiebig and others to describe when
systems have symbolic extensions and their entropy properties.
Combining this development with earlier work of Buzzi (and Yomdin) one
obtains the theorem that every $C^{\infty}$
diffeomorphism possesses so-called principal symbolic
extensions. These are symbolic extensions such that the projection
maps preserve entropy for every invariant probability measure. We
describe results for systems with low smoothness obtained recently
with Tomasz Downarowicz. In particular, a $C^1$ generic non-Anosov
surface diffeomorphism has no symbolic extension at all.

**
Nicholas Ormes**,
University of Denver
*
Topological Realization of Families of Ergodic Systems
*

This is joint work with Isaac Kornfeld. We will report on
progress on the following problem: given a family
$\{(T_{\alpha},\nu_{\alpha}): \alpha \in A\}$ of ergodic automorphisms
of non-atomic Lebesgue probability spaces, find a single minimal
homeomorphism $S$ of the Cantor set such that the set of ergodic
$S$-invariant Borel measures is $\{\mu_{\alpha} : \alpha \in A\}$ where
$(S,\mu_{\alpha})$ is measurably conjugate to
$(T_{\alpha},\nu_{\alpha})$. In this talk we will show that this can be
done when $A$ is finite. As far as we know, this was an open question
even in the special case of a family consisting of 2 arbitrary
irrational rotations.

For arbitrary collections, our technique along with some additional
ingredients seems to give the following: if it is possible to construct
an aperiodic, not necessarily minimal, $S$ as above then it is possible
to construct a minimal $S$. Furthermore, it follows that we are able to
achieve our realization within the topological orbit equivalence class
of any minimal system where the space of invariant measures is affinely
homeomorphic to the space of invariant measures for the non-minimal
action.

**
Mark Pollicott**,
University of Manchester, England
*
The Dimension of Fat Sierpinski Carpets
*

The Sierpinski carpet and the Sierpinski triangle are familiar
examples of self-similar fractals and their dimension is particularly
easy to calculate. We want to consider a simple modification of this
construction, in which there are overlaps, and the Hausdorff dimension
of the resulting set. This is joint work with my
graduate student Thomas
Jordan.

**
Anthony Quas, **
University of Memphis
*
Duality in the return-time theorem
*

The return time theorem states that for any system (X,B,\mu,T) and any
f in L^p(X), there is a subset X_0 of measure 1 such that for any
system (Y,F,\nu,S) and any g xin L^q(Y), where q is the conjugate
exponent to p, the averages

(1/N)[f(Tx)g(Sy) + f(T^2x)g(S^2y) + ... + f(T^Nx)g(S^Ny)]

converge
for ALL x\in X_0 for almost every y in Y. If a theorem of this type
holds when 1/p + 1/q > 1, then there is said to be duality-breaking. We
discuss recent results of Assani, Buczolich and Mauldin showing that in
the case where p=q=1, there is no duality breaking.

**
Monica Moreno Rocha, **
Tufts University
*
Rational Maps with generalized Sierpinski gaskets
as Julia sets
*

In this talk we describe the dynamics of a class
of rational maps with generalized Sierpinski gaskets as Julia
sets. The well-known Sierpinski triangle will be our starting
example to illustrate the definition of a generalized gasket.
Then we will provide a description of the dynamics of the rational
family $z \to z^2+\lambda / z^2$ in the Riemann sphere for Misiurewicz
values of the complex parameter $\lambda$. Using this information,
we will present a topological model of the Julia set to show when two
generalized Sierpinski Julia sets are not homeomorphic.
This is joint work with R.L. Devaney (Boston University)
and S. Siegmund (University of Frankfurt).

**
Rafael Ruggiero**,
PUC-Rio, Brazil
*
Rigidity of surfaces admitting a C^2, codimension one foliation
invariant by the geodesic flow.
*

Let (M,g) be a closed, orientable surface. If the geodesic flow preserves
a C^2, codimension one foliation, then the Gaussian curvature is either
equal to zero or equal to a negative constant.

**
Evelyn Sander**,
George Mason University
*
Crossing Bifurcations
and Unstable Dimension Variability
*

A crisis is a global bifurcation in which a chaotic attractor has a
discontinuous change in size or suddenly disappears as a scalar parameter
of the system is varied. Examples of crises for two-dimensional maps
occur simultaneously with tangencies of stable and unstable manifolds of
underlying saddle orbits. In this talk, we will describe a different type
of global bifurcation which can result in a crisis. This bifurcation does
not involve a tangency and cannot occur in maps of dimension smaller than
three. An important distinction in the type of global bifurcation is made,
depending on whether the crossing invariant manifolds are twisted or not.
We introduce this new concept by presenting an example of a parametrized
three-dimensional chaotic attractor which undergoes a crisis at a crossing
bifurcation with twisted manifolds. The crisis also produces unstable
dimension variability.

**
Maria Saprykina **,
Royal Institute of Technology, Sweden
*
New examples in ergodic theory
*

I shall speak about the result of my joint work with B. Fayad.
We present a construction method providing area preserving weakly mixing
diffeomorphisms on a manifold M equal to the torus $\mathbb T^d$,
annulus $A=\mathbb T\times [0,1]$ or disc $\mathbb D^2=\{x^2+y^2\leq 1\}$.

We denote by $S_t$ the elements of (a particular)
circle action on $M$.
For any Liouville number $\alpha$ we construct a
sequence of area-preserving diffeomorphisms
$H_n$ such that the sequence $H_n\circ S_\a\circ H_n^{-1}$
converges to a smooth weakly mixing diffeomorphism of $M$. The method is a
quantitative version of the approximation by conjugations
construction introduced by D.Anosov and A.Katok in the 70-th.

For $M=A$ or $\mathbb D^2$, this result proves the following dichotomy:
$\alpha \in \RR \setminus\mathbb Q$ is Diophantine if and only if
there is no ergodic diffeomorphism of $M$
whose rotation number on the boundary equals $\alpha$.
One part of the dichotomy follows from our constructions, the other is
an unpublished result of Michael
Herman asserting that if $\alpha$ is Diophantine, then any area preserving
diffeomorphism
with rotation number $\alpha$ on the boundary displays smooth
invariant curves arbitrarily close to the boundary which clearly
precludes ergodicity or even topological transitivity.

On the torus our method gives explicit approximation by conjugations
constructions, providing real analytic weakly mixing diffeomorphisms.

**
Omri Sarig**,
Penn State University
*
Invariant Measures for Horocycle Flows on Abelian Covers
*

Furstenberg showed that the horocycle flow on a compact
hyperbolic surface has exactly one invariant probability measure. The
horocycle flow on a Z^d-cover of a compact hyperbolic surface has no
finite invariant measures at all (Ratner), and this raises the question of
infinite invariant measures. Babillot & Ledrappier constructed a
d-parameter family of infinite ergodic invariant measures which are Radon:
compact sets have finite measure. They then conjectured that their family
contains all ergodic invariant Radon measures (up to a constant
multiple). I will present a proof of this.

**
Anna Talitskaya**,
Penn State University

(PhD student, advised by
Yakov Pesin)
*
Construction of a hyperbolic Bernoulli flow on any manifold
*

**
Klaus Thomsen**,
Aarhus University, Denmark
*
The derived shift space of a beta-shift
*

The derived shift space is a key ingredient in the structure
of a sofic shift space, and it makes sense much more generally. To
understand its role in non-sofic subshifts, it is natural
to seek to describe it for the most familiar (non-sofic) synchronized
shift spaces. It turns out that to decide which shift spaces can occur as the
derived shift space of a beta-shift, one is confronted with the following
language theoretic problem which was formulated in computer science a
decade ago: Which languages can be realized as the finite words
occurring infinitely often
in an infinite word? I will describe the answer to this question, and
explain how it leads to the following conclusion: A shift space is the
derived shift space of a beta-shift if and only if it is chain-transitive.

**
Marius Urbanski**,
University of North Texas
*
The dynamics of elliptic functions
*

The lower bound for the Hausdorff dimension of the Julia sets of
elliptic functions will be provided. The Hausdorff and packing measures
for critically non-recurrent elliptic functions will be discussed. The
appropriate Gibbs states will be argued to exist.

**
Howard Weiss**,
Penn State University
*
The Remarkable Dynamics of a Nonlinear Age-Structured Population
Model
*

All age-structured population forecasting, for both humans and
animals, is done using the LINEAR Leslie model. Perhaps not coincidently,
the 2000 US census showed that the best demographic models based on 1990
census data underestimated the U.S. population by six million people. Many
demographers, population biologists, and ecologists are now looking to
nonlinear models for more accurate population models and projections.

The linear Leslie model contains two parameters (constants!) for each
``generation'': the per-capita fertility rate and the survival probability
(of surviving into the next generation). For over 20 years some population
experts have advocated extending the linear Leslie model to allow these
parameters to depend on population size and perhaps time, and there have
been a small handful of papers indicating the existence of complicated
dynamics for some nonlinear models.

We have started a project to systematically study the global dynamics
and bifurcations for nonlinear Leslie models where the fertility rates
and survival probabilities have various natural forms.

In this talk I will discuss the dynamics of an overcompensatory Leslie
model where the fertility rates decay exponentially with population size.
This model is ``highlighted'' in the new edition of Caswell's treatise on
population models. We find a plethora of remarkably complicated dynamical
behaviors, many of which have not been previously observed in structured
population models, and which may give rise to new paradigms in population
biology and ecology. In particular, in the two and three generation
models we found: period doubling cascades, attracting closed curves which
bifurcate into strange attractors, various routes to chaos, multiple
co-existing strange attractors with large basins, three types of crises --
which can cause a discontinuous large population swing, merging of
attractors, phase locking, and transient chaos. We also found one
parameter families that exhibit most of these phenomena.

Along the way we found (and explained) two different bifurcation cascades
transforming an attracting invariant closed curve into a strange
attractor. Finally, we explicitly showed that some of the more exotic
phenomena arise from homoclinic tangencies.

**
Yingfei Yi **,
Georgia Institute of Technology
*
On almost automorphic dynamics
*

Almost automorphy is a notion first introduced by S. Bochner
in 1955 to generalize the almost periodic one. It is proven to
be a fundamental notion in characterizing
multi-frequency phenomena and their generating dynamical complexity.
This lecture will give a survey of the topic along with
some discussions on related
problems arising in dynamical systems and differential equations.