on Dynamical Systems and Related Topics

University of Maryland, March 17-20, 2001

**
Karen Ball,
**
University of Maryland
*
Skew products which are not standard
*

Abstract form:
dvi or
pdf.

**
Henk Bruin,
**
University of Groningen
*
Expansion of derivatives and the measure of Julia sets
*

This is a report on joint work with Sebastian van Strien
(University of Warwick). Given a rational map on the
Riemann sphere, we show that summable growth rates of
the derivatives along the critical orbits imply
similar growth rates for every non-precritical orbit
in the Julia set.
This is used to prove that a rational map satisfying
such condition has a Julia set which is either the whole
sphere, or has Lebesgue measure zero.

**
William Cowieson,
**
New York University
*
Stochastic stability for invariant densities
of piecewise expanding maps
*

Under suitable regularity conditions, "most"
piecewise expanding maps T of a d-dimensional domain
have an invariant measure which is absolutely continuous
w.r.t. Lebesgue measure. Moreover, this holds also for
suitable random perturbations of T, and the stationary
distributions obtained tend to invariant densities of T
as the noise tends to 0.

**
Beverly Diamond,
**
College of Charleston
*
A complete invariant for the topology of 1-d substitution tiling spaces
*

This talk is based on joint work with Marcy Barge (Montana State
University). Given a primitive, nonperiodic substitution $\varphi$, the
associated tiling space ${\mathcal T}_\varphi$ has a finite (nonzero)
number of asymptotic composants. We describe the form of these asymptotic
composants and give a bound for their number. We make use of this
description to discuss the topology of tiling spaces and provide examples
indicating that for substitution minimal systems, flow equivalence and
orbit equivalence are independent.

**
Filiz Dogru,
**
Pennsylvania State University
*
On polygonal dual billiards in the hyperbolic plane
*

The dynamics of the polygonal dual billiard map
depends on the size and shape of the respective
polygon. For a class of polygons, called "large",
all orbits of the dual billiard map escape to infinity
(in sharp contrast with the Euclidean case).

**
Dmitry Dolgopyat,
**
Penn State University
*
Differentiation of SRB measures
*

We discuss how invariant measures, Invariant plane fields,
Lyapunov exponents etc. of dynamical systems depend on parameters
in the class of partially hyperbolic sistems.

**
John Franks,
**
Northwestern University
*
Group actions on one-manifolds
*

(Mathematics Department Colloquium Talk, Friday March 16, 3 p.m.)

**
Kresimir Josic,
**
Boston University
*
The Structure of the Julia Set of Stable Exponentials
*

We consider the dynamics of the complex exponential
$\lambda e^z$, where $\lambda$ is chosen so that the dynamical
system defined by the map has an attracting periodic orbit.
If the attracting orbit consists of a fixed point, the Julia
set for this map is a Cantor bouquet and it has been well studied.
We show that in the case the period of the attracting orbit is greater than
one a set similar to a Cantor bouquet still exists, but some of the
hairs comprising the bouquet are pinched together. The
structure of this set and the accessibility of the points within it
will be discussed.

**
Vadim Kaloshin,
**
New York University
*
Dynamics of an Oil Spot on
the Surface of Ocean and Dynamics with
Nonzero Lyapunov Exponents
*

Consider a surface of ocean (2-dimensional pane)
and shade a unit disk in the origin (an oil spot or
a passive scalar). Evolition of the surface of ocean
in time is described by an area preserving Stochastic
Differential Equation. What is an assymptotic behavour
of this Oil Spot? What is its diameter for large time?
and so on.
It turns out that different features of evolution of
such a spot generically can be described using methods
of systems with nonzero Lyapunov exponents.
This will be report on a joint work with Dima
Dolgopyat and Lenya Koralov.

**
Bryna Kra,
**
Penn State University
*
Some non-conventional ergodic averages
*

I will discuss recent results on non-conventional ergodic averages,
including products of functions and IP-like expressions. For such
averages, the limiting function is, in general, not constant but is
determined by a factor. This is joint work with Bernard Host.

**
Francois Ledrappier,
**
CNRS and Northwestern University
*
A weak Besicovich property and applications
*

The Besicovich property is a strong tool of geometric measure theory.
When computing or estimating Hausdorff dimensions, the full strength of the
property is not needed. We propose a weaker property and present (mostly
future) applications.

**
Elon Lindenstrauss,
**
Institute for Advanced Study
*
p-adic foliations and equidistribution
*

I will consider a very simple dynamical system, namely the
1-torus $R/Z $ with the map $x \mapsto mx mod 1 $, and discuss some
long-standing conjectures regarding this map.
I will also show how one can use $p $-adic analogues of results from the
theory of multidimensional smooth dynamical systems in this context, and in
particular, we show that if $\mu $ is ergodic under the times $m $ map and
has positive entropy, then $\mu $-a.s. $\{r ^n x\} $ is equidistributed as
long as $m$ does not divide any power of $r $ (this was previously known
only if $r $ and $m $ are relatively prime).

**
Brian Martensen,
**
Montana State University
*
The local topological complexity of C^r-diffeomorphisms near
homoclinic tangency
*

If F is a C^r-diffeomorphism with periodic saddle p so that a
branch of the unstable manifold of p has a tangency with the stable
manifold, then C^r-close to F is a diffeomorphism such that the closure of
this branch of the unstable manifold everywhere locally contains all
chainable continua.

**
Alica Miller,
**
Michigan State University
*
Minimality of restrictions, group-extensions and
products of compact minimal abelian flows
*

Abstract form:
dvi or
pdf.

**
Linda Moniz,
**
University of Maryland
*
Upper Bound Sets for Dynamical Behavior
*

We describe an upper bound method for finding periodic points and
other relevant features of $C^2$ maps in the plane. This method
produces a set D that provides necessary conditions for dynamical
behavior, i.e. if a point exhibits the behavior (e.g. a periodic point
of period k) it must be in the set D. In many cases we can find a
nested sequence of sets D that converges to the set of points
with the desired property.

**
Nicholas S. Ormes,
**
University of Connecticut
*
Symbolic Dynamics and Ordered Cohomology.
*

The speaker will discuss the use of ordered cohomology
groups
as invariants for symbolic dynamical systems. Various authors (e.g.
Boyle/Handelman, Giordano/Putnam/Skau) have used the order structure
to
form a stronger invariant than the cohomology groups alone. More
recent
results (e.g. Ormes/Radin/Sadun, Yi) seem to suggest that the
invariance
of certain ordered cohomology groups is a purely topological
phenomenon,
not relying on underlying dynamics.

**
Dhanurjay (DJ) Patil
**
University of Maryland
*
Local low dimensionality of atmospheric dynamics -- when good
forecasts go bad
*

From the dynamical systems point of view, the behavior of the
Earth's atmosphere is extremely high dimensional (e.g., a
realistic atmospheric model based on a modal expansion would
necessarily include many modes). In spite of the atmosphere's
high dimensionality, in this talk I will demonstrate that, in a
suitable sense, the local finite-time atmospheric dynamics is
often low dimensional. Furthermore, I will show how this finding
has important implications for weather forecasting. More
generally, this behavior may be common to other physical
spatio-temporally chaotic systems, and these systems may also be
amenable to the type of analysis that is introduced for the
atmosphere.

**
Yakov Pesin,
**
Penn State University
*
On the Existence of Bernoulli Diffeomorphisms With Nonzero
Lyapunov Exponents
*

We show that a smooth compact Riemannian manifold of
dimension $\ge 2$ admits a Bernoulli diffeomorphism with
nonzero Lyapunov exponents.

**
Karl Petersen,
**
University of North Carolina
*
From K to Super-K
*

The future tail field of a finite-state stationary process consists of
all
the information that remains after time has run its course; for
Kolmogorov
processes, this tail field is trivial. The future fine tail field
consists
of the residual information in case a limited amount of record
keeping,
namely symbol counts accumulated from time 0 onward, is allowed; for
Bernoulli processes the future fine tail field is also trivial, but it
need not be trivial for other partitions that generate the process.
With
J.-P. Thouvenot, we have been working to recode Kolmogorov processes
in
such a way that the future fine tail field is trivial. We are able to
accomplish such a recoding in case there is a direct Bernoulli factor
by
using the asymptotic flatness of the distribution of symbol counts in
Bernoulli processes.

**
Mason Porter,
**
Cornell University
*
Quantum Chaos in Vibrating Billiard Systems
*

We consider quantum billiards with vibrating surfaces, which can be
used
as a semiquantal description of nonadiabatic coupling in polyatomic
molecules and nanomechanical devices. We review the various notions
of
quantum chaos, focusing on that observed in semiquantal systems. We
derive necessary conditions for chaotic behavior in quantum billiards
with
a single degree-of-vibration, and we discuss the relationship between
the
observed quantum chaos (which is known as 'semiquantum chaos') and
classical Hamiltonian chaos. The present analysis is illustrated
by considering the radially vibrating spherical quantum billiard.

**
Anthony Quas,
**
University of Memphis
*
Convexity, Random Tilings and Shifts of Finite type
*

In one dimension, it is known that a mixing shift of finite type
has a unique measure of maximal entropy and this measure is isomorphic to
a Bernoulli shift. This fails badly in higher dimensions. In this talk, we
find a class of higher-dimensional shifts of finite type for which there
are Bernoulli measures with entropy arbitrarily close to the topological
entropy - this can be viewed as a strong variational principle for these
systems. On the way, we establish some general facts about
higher-dimensional entropy.

**
Rafael Oswaldo Ruggiero,
**
Pontificia Universidade Catolica do Rio de Janeiro
*
On the divergence of geodesic rays in manifolds without conjugate
points
*

Let $(M,g)$ be a compact Riemannian manifold without conjugate points. Denote
by $H(p,v)$ the horosphere
containing $p$, in the universal covering $\tilde{M}$, of the geodesic
$\gamma_{(p,v)}(t)$
(defined by $\gamma_{(p,v)}(0)=p$,
$\gamma_{(p,v)}'(0)=v$). Suposse that the map \( (p,v) \longrightarrow H(p,v)\)
is continuous
(uniformly on compact subsets of $\tilde{M}$). Then we prove that the distance
$d(\gamma(t),\beta(t))$ between two different
geodesic rays starting at any point $q \in \tilde{M}$ tends to infinity with
$t$. We use this fact
to show that a continuous local product structure of horospheres in $T_{1}M$
implies that the
geodesic flow is expansive, and hence $\tilde{M}$ is a Gromov
hyperbolic space.

This result can be viewed as a topological counterpart of a wellknown
theorem of P. Eberlein which says that the Anosov property is equivalent to the
transversality
of Green bundles.

**
Martin Sambarino,
**
University of Maryland
*
On the existence of homoclinic orbits
*

We will show that there exists an open and dense set ${\cal{U}}$ in
$Diff^1(M)$,
where $M$ is any compact manifold, such that each $f\in{\cal{U}}$
either is a Morse-Smale
diffeomorphism or exhibits a transverse homoclinic orbit.

**
Lai-Sang Young,
**
New York University
*
From invariant curves to strange attractors
*

Simple mechanical systems, when subjected to external
periodic forcing, can exhibit a rich array of dynamical behaviors
as parameters are varied. I will discuss among other things the
existence of global strange attractors with fully stochastic
properties for a class of second order ODEs.

**
Michiko Yuri,
**
Sapporo University
*
Weak Gibbs measures for
potentials of weak bounded variation and subexponential
instability
*

Abstract form:
dvi or
pdf.

**
Piotr Zgliczynski,
**
Indiana University, Bloomington
*
Isolating segments and chaotic behavior.
*

Using topological tools (isolating segments) we prove that the
differential equation on the plane
$ z'=\bar{z}(1+|z|^2 e^{i\kappa t })$
has a symbolic dynamics on three symbols for $0 \kappa \leq 0.495$.
We show also an existence of an infinite number of geometrically
distinct
solutions, which are homoclinic to the constant zero solution.