on Dynamical Systems and Related Topics

University of Maryland, March 23-26, 2002

**
Ethan Akin**,
City College of New York (CUNY)
*
Distality Concepts and Ellis Actions
*

Following some ideas of Blanchard et al. we look at some
generalizations of distality using Ellis semigroups to unify the
picture.
The most general of these, semi-distal systems are closed under
distal and asymptotic lifts and minimal semi-distal systems are
disjoint from all weak mixing minimal systems.

**
Idris Assani,
**
University of North Carolina, Chapel Hill
*
On the convergence of the averages
$\frac{1}{N}\sum_{n=1}^Nf(S^nx)g(T^nx)$
*

Motivated by questions of H. Furstenberg we study the a.e convergence
of the averages $\frac{1}{N}\sum_{n=1}^Nf(S^nx)g(T^nx)$ where T and S are
two commuting measure preserving transformations. We also consider the
convergence of the averages $\frac{1}{N}\sum_{n=1}^N T^n(gS^nf)$ where T
and S are commuting operators and g a $L^{\infty}$ function.

**
Joe Auslander,
**
University of Maryland, College Park
*
The proximal relation in topological dynamics
*

**
Karen Ball,
**
University of Maryland, College Park
*
Entropy and random walks on random sceneries
*

View the abstract in
dvi or
pdf format.

**
Vitaly Bergelson,
**
Ohio State University
*
Multiple recurrence and the properties of large sets *
(Friday Math Department Colloquium)

Many familiar theorems in various areas of mathematics have the following common feature: if A is a large set, then the set of its differences (or, sometimes, the set of distances between its elements) is VERY large. For example:

- If A is a set of reals having positive Lebesgue measure, then there exists a positive real a, so that A-A contains the interval (-a,a).
- If A is a set of natural numbers having positive upper density, then for any polynomial p(n) having integer coefficients and zero constant term, the set A-A contains infinitely many integers of the form p(n).
- If F is an infinite algebraic field and G is a subgroup of finite index in the multiplicative group F*, then G-G = F.

The talk is intended for a general audience.

**
Vitaly Bergelson,
**
Ohio State University
*
Towards a non-commutative Hindman's Theorem
*

The celebrated Hindman theorem states that for any finite partition of
an infinite Abelian (semi-)group, one of the cells of the partition
contains an infinite set S, such that all finite products of distinct
elements from S belong to the same cell. In our talk we shall formulate
some conjectures dealing with possible non-commutative extensions of
Hindman's theorem and will present recent results (obtained with the
help of topological and measurable dynamics) which support these
conjectures.

**
Sergey Bezuglyi,
**
University of New South Wales and Institute for Low Temperature
Physics, Kharkov, Ukraine
*
Weak and uniform topologies for Cantor minimal systems *

The talk is based on two papers
with Jan Kwiatkowski (click for
postscript version):
"The topological full group of a Cantor
minimal system is dense in the full group"
and
"Topologies on full groups and normalizers
of Cantor minimal systems".

**
Francois Blanchard,
** IML-CNRS, Marseilles
*
Cellular automata and the Besicovitch topology on the 2-shift *

Instead of endowing $X = (0,1)^{\bf Z}$ with the usual product
topology, one can consider the shift-invariant Besicovich pseudo-metric
$$ d_B(x,y) = \limsup_{l\to \infty} {\#\{j \in [-l,l]:\; x_j \neq y_j\}\over
2l+1}. $$
It was introduced by Cattaneo, Formenti, Mazoyer and Margara to
answer the claim, widespread
among physicists and computer scientists, that, considering the full
shift as a cellular
automaton, it is not heuristically very different from the identity CA because
it does not
change the symbols, only their positions.
Observe that $d_B$ is not a metric because $d_B(x,y)=0$ for some
$x\ne y$. The quotient metric
space
$X/d_B$ is not compact, not separated but complete. Usual cellular automata act
continuously on this space. The comparison of the dynamics of one
given CA $F$ on $X$ and on
$X/d_B$ is
very instructive. On $X/d_B$ the shift CA becomes an isometry, and in general
$F$ has less `chaoticity' with respect to
$d_B$ than in the usual topology. One striking instance is that no CA can be
topologically transitive for $d_B$.

**
Leo Butler,
**
Northwestern University
*
Integrable hamiltonian systems with positive
Lebesgue metric entropy
*

A completely integrable flow is a flow whose phase space
contains an open dense set fibred by invariant tori,
and the flow on these tori is a translation-type flow.
A common method to demonstrate the real-analytic non-integrability
of a hamiltonian flow has been to demonstrate the existence
of a subsystem that is incompatible with real-analytic
integrability.

In this talk, it is shown that any real-analytic symplectic map can be
embedded as a subsystem of a time-T map
of a smoothly integrable,
real-analytic hamiltonian flow. We also construct
a smoothly integrable hamiltonian flow on a Poisson
manifold that preserves a smooth positive measure
with respect to which the time one map has positive entropy.

For the entire paper, click to
www.math.northwestern.edu/~lbutler/research.

**
Toke Carlsen
**, University of Copenhagen and University of Maryland
*
Invariants for substitutional dynamical systems
*

I will define an invariant for substitutional dynamical systems and
show an explicit and algorithmic way to compute this invariant. This
invariant is based on the K-theory of the so-called Matsumoto
C*-algebra. I will also discuss how this invariant is related to the
dimension group Durand, Host and Skau have studied.

**
Tomasz Downarowicz,
**
Wroclaw University of Technology and Michigan State University
*
Sex entropy: symbolic extensions and the entropy structure of a
dynamical system
*

In a topological dynamical system (unless it is expansive) the
entropy of an invariant measure does inform us about the scale
(or resolution) at which a given part of that entropy can be detected.
As illustration imagine a system consisting of infinitely many copies
of the full two-shift decreasing in size and accumulating at a fixpoint.
This leads to certain difficulties in encoding the action symbolically,
i.e., in building a symbolic extension. Such extension (being expansive)
requires that all of entropy must become detectable in one fixed
scale (the expansive constant), in other words all local chaos must
be "enlarged" to the same size. This may result in significant
increase of the overall topological entropy. We will propose an
approach which allows to control the entropy and its scale for all
invariant measures and hence to compute the symbolic extension
(abbreviation: sex) entropy in topological dynamics.

Let T be a continuous selfmap of a compact metrizable space X.
Consider a refining sequence of finite Borel partitions P_n,
where an element of P_n has diameter at most 1/n and has
boundary of measure zero for any nonatomic T-invariant probability.
Define functions on the space M_T of T-invariant Borel probabilities,

h_n: M_T --> R

m |--> h_m(T,P_n).

The sequence (h_n) is an * entropy sequence * for T.
(There is a more general way of defining entropy sequences
which we will skip.) This sequence allows us to use separation
theorems and semicontinuity properties in functional analysis.

Given a sequence of nonnegative upper semicontinuous functions
f_n on a Choquet simplex, define a * super envelope*
of the sequence to be a function E such that each difference
E-f_n is nonnegative and u.s.c.

THEOREM
E is an affine super envelope for the entropy sequence h_n
<==> there exists a quotient map p from a subshift S onto T
such that

E(m) = max {h_{mu}(S): mu is in M_S and p(mu) = m}.

The infimum of all affine super envelopes is again a superenvelope.
This mimimal superenvelope describes at each measure the inf of
the entropy jumps to any preimage measure in a symbolic extension.
The theorem then leads to examples of subtle phenomena (e.g. nonattainment
of sex entropy by any extension and
failure of "ergodic decomposition" for sex entropy) and
also to a definition
of the entropy structure of a system, which comes from an
appropriate equivalence relation on entropy sequences.

This is joint work with Mike Boyle.

**
David Duncan
**
*
A Wiener-Wintner Double Recurrence Theorem
View the abstract in
dvi or
pdf format.
*

**
Manfred Einsiedler
**
Pennsylvania State University
*
Invariant measures on SL(n,R)/Gamma
*

View the abstract in
dvi or
pdf format.

**
Bassam Fayad**,
CNRS and Pennsylania State University
*
On the multiplicative cohomological equation E_{\lambda}:
g(x+\alpha) = g(x) e^{2\pi i \lambda \phi(x)}
*

For irrational numbers \alpha and smooth functions \phi, we study the existence of measurable solutions for the equations E_{\lambda}, \lambda \in {\R}^*.

This problem is directly related to the dynamics of the special flow over R_\alpha and under \phi as well as to the skew products on the torus

(x,y) \rightarrow (x+ \alpha, y+ \phi(x)).

**
Matthew Foreman**,
The Classification of Ergodic Measure
Preserving Systems
*
University of California, Irvine
*

In this talk we describe a method of attacking
the isomorphism problem for measure preserving
systems. Using the tools of descriptive set
theory (in particular Borel Reduction) one
can determine the inherent difficulty in
classifying measure preserving systems. Examples
are given in the talk of positive classification
results (for distal transformations) as well as
new unclassifiability results for the whole
class of ergodic transformations.

**
Nikos Frantzikinakis
**
Stanford University
*
The structure of strongly stationary systems
*

View the abstract in
dvi or
pdf format.

**
Eli Glasner,
**
Tel Aviv University
*
Universal minimal dynamical systems
*

View the abstract in
dvi or
pdf format.

**
Jonathan King
**,
University of Florida, Gainesville
*
Joining-closure and genericity
*

**
Andrey Kochergin**,
Moscow State University
*
The mixing "almost Lipshitz" reparametrization of a
flow on the two-dimensional torus
*

For any regularity condition on continuity
which is weaker than Lipshitz, there exists
an irrational linear flow and a time change
satisfying the regularity condition, such
that the time-changed flow is mixing.

**
Chao-Hui Lin,
**
University of Maryland
*
Kakutani shift equivalence for uniformly dyadic
endomorphisms
*

**
Doug Lind,
**
University of Washington, Seattle
*
Algebraic Z^d-actions of rank one
*

In joint work with Einsiedler, we give a number of
different characterizations of algebraic Z^d-actions of rank
one, i.e. those for which every element has finite entropy.
A common feature is the existence of "eigenspaces" for such
actions, even in totally disconnected examples like
Ledrappier's. Generalizing work of Marcus and Newhouse, we
compute the topological entropy of a skew product with base
a shift of finite type and fiber maps coming from a rank one
algebraic Z^d-action. The answer can be expressed as the
largest of a finite number of topological pressures, each of
which can be computed explicitly. This result allows
computation of Friedland's relational entropy for any pair
of commuting group automorphisms, disproving a conjecture of
Geller and Pollicott.

**
Nelson Markley,
**
Lehigh University
*
Remote limit points on surfaces
*

View the abstract in
dvi or
pdf format.

**
Xavier Mela
**,
University of North Carolina, Chapel Hill
*
Measurable and topological properties of some nonstationary adic
transformations
*

We describe a class of nonstationary adic transformations in
terms of their Bratelli diagrams, and by their cutting-and-stacking
equivalents. These transformations are isomorphic to substitution
subshifts defined by countably many substitutions. We discuss properties
such as topological weak mixing, growth of the complexity function, rank,
and the loosely Bernoulli property.

**
Mahesh Nerurkar,
**
Rutgers at Camden
*
On Characterizing Distal Points
*

We present an extension (to arbitrary acting group) of a result
of H. Furstenberg and J. Auslander characterizing distal points as those
that are product recurrent. This result is a consequence of density of
maximal idempotents. The density result is proved using the
``rarification'' of the construction used in the proof of Galvin's
theorem.

**
Yakov Pesin,
**
Pennsylvania State University
*
Stable ergodicity and Lyapunov exponents
*

I present some results on stable
ergodicity of partially hyperbolic diffeomorphisms with non-zero
Lyapunov exponents. The main tool is local ergodicity theory
for non-uniformly hyperbolic systems.

**
Karl Petersen,
**
University of North Carolina, Chapel Hill
*
Bounding the number of measures of maximal entropy *

View the abstract in
dvi or
pdf format.

**
Ian Putnam,
**
University of Victoria
*
Recent progress on topological orbit equivalence
*

I will describe some recent work with Giordano (Ottawa) and
Skau (Trondheim) on topological orbit equivalence for dynamics on Cantor
sets. I will begin with an introduction and a new look at some of our
earlier results on actions of a single minimal homeomorphism and so-called
AF-relations. I will describe recent results for minimal Z^2 actions.

**
Charles Radin,
**
University of Texas, Austin
*
Using an ergodic theorem of Nevo to control problems of optimal
density
*

The old problem of determining the optimal density for packings of
equal spheres in Euclidean n-space is well defined but computationally
difficult for dimensions 3 or higher.

In hyperbolic spaces there is an added complication, as it has been
accepted for many years that the very notion of optimal density is too
difficult to define, as demonstrated by numerous examples. We will
describe how a recent pointwise ergodic theorem of Alex Nevo et al
allows one to control those packings with ill defined density via sets
of measure zero with respect to probability measures invariant under
the congruence group, the measures being used both as a conceptual and
computational tool.

For a preprint by L. Bowen and CR, click
here.

**
Pierre-Paul Romagnoli**, Universidad de Chile
*
Measure Theoretical Entropy for Covers and a Local
Variational Principle
*

We give two new notions
of measure theoretical and topological entropy for
open covers,
that extend the classical concept of measure
theoretical entropy of measurable
partitions. Using these notions we prove a local
variational principle for a fixed open
cover. We give some
applications to compute topological entropy, measure
theoretical
entropy pairs and entropy of induced systems. Some
computations of these
new concepts for subshifts of finite type are also
given.

**
Ayse Sahin**,
DePaul University
*
Entropy and locally maximal entropy in Z^d subshifts
*

We discuss the behavior
of the entropy function and mixing properties of high-entropy
subshifts of higher dimensional shifts of finite type
(SFTs). We provide examples illustrating the necessity of strong
topological mixing hypotheses in existing embedding and representation
theorems for Z^d systems. The talk is based on the joint work
with Anthony Quas,
"Entropy gaps and locally maximal entropy in Z^d subshifts"; for
the postscript file of this paper, click
here.

**
Howard Weiss,
**
Pennsylvania State University
*
Does the free energy determine the potential for a lattice
gas? ("Can you hear the shape of a cookie cutter?")
*

(Joint with
Mark
Pollictt)
The lattice gas provides an
important and illuminating family of models in statistical physics. An
interaction on a lattice in Z^r determines an
idealized lattice gas system with a potential. The pressure
and free energy
are fundamental characteristics of the system. However, even for
the simpliest lattice systems, the information about the potential that
the free energy captures is subtle and poorly understood. We study
whether, or to what extent, H\"older continuous potentials for certain
model systems are determined by their free energy.

In the language of dynamical systems, we study whether a H\"older
continuous potential for a subshift of finite type or cookie cutter is
naturally determined by its unmarked periodic orbit spectrum, beta
function (essentially the free energy), or zeta function. It turns out
that these problems have striking analogies to fascinating questions in
spectral geometry.

**
Huang Wen,
**
University of
science and technology of china, Hefei
*
Topological K-system, a third approach
View the abstract in
dvi or
pdf format.
*

**
Alistair Windsor**,
Pennsylvania State University
*
A Weak Mixing Dichotomy for Time Changes of Linear Flows
*

We show that, for a class of functions exhibiting regular
decay of Fourier coefficients, the special flow over a rotation and
under the function is either weak mixing or conjugate to the linear
flow. This dichotomy coexists with our earlier result which showed
that for certain Liouville rotations there are analytic functions
such that the special flow over the rotation and under the function
exhibits mixed spectrum (and in particular is neither weak mixing nor
conjugate to a linear flow). The condition of rapid decay of Fourier
coefficients means that for each rotation and each cohomology class
we are able to isolate a particularly nice representative and
calculate with this.

This result is joint work with Bassam Fayad (CNRS, France) and
Anatole Katok (Penn State).

**
Xiangdong Ye,
**
University of Science and Technology of China, Hefei
*
Transitive systems with zero sequence entropy and
sequence entropy pairs
*

A measure-preserving transformation (resp. a topological system) is null
if the metric (resp. topological) sequence entropy is zero for any sequence.
Kushnirenko showed that an ergodic measure-preserving
transformation T has discrete spectrum if and only if it is null.
We prove that for a minimal
system the above statement remains true modulo an almost one-to-one
extension, i.e. if a minimal system (X,T) is null, then (X,T) is
an almost one-to-one extension of an equicontinuous system.
It allows us to show that
a scattering system is disjoint from any null minimal system.
Moreover, we show that if a transitive non-minimal system (X,T) is null
then the access time N(U,V) has zero upper Banach density.
Examples of null minimal systems which are not equicontinuous exist.

Localizing the notion of sequence entropy,
we define sequence entropy pairs and show that there is a maximal null
factor for any system. Meanwhile, we define a weaker notion, namely
weak mixing pairs. It turns out
that a system is weakly mixing if and only if any pair not in the
diagonal is a sequence entropy pair if and only if the same holds for
a weak mixing pair, answering Question 3.9 in [BHM] by Blanchard, Host
and Maass.
For a group action we show that the factor induced by the smallest
invariant equivalence relation containing weak mixing pairs
is equicontinuous, supplying another proof concerning regionally
proximal relation. Furthermore, for a minimal distal
system the set of sequence entropy pairs coincides with the regionally
proximal relation and thus a non-equicontinuous minimal distal system
is not null.

**
Jim Yorke **,
University of Maryland
*
Learning about reality from observation
*

Takens, Ruelle, Eckmann, Sano, and Sawada launched an
investigation of images of attractors of dynamical systems. If A is a
compact invariant set for a map f on R^n and g:R^n -> R^m where n > m is a
"typical" smooth map, when can we say A and g(A) are similar, based only on
knowledge of the images in R^m of trajectories in A? For example, under
what conditions on g(A) (and the induced dynamics thereon) are A and g(A)
homeomorphic? Are their Lyapunov exponents the same? Or, more precisely,
which of their Lyapunov exponents are the same? This talk addresses these
questions.

In answering these questions, a fundamental problem arises about an
arbitrary compact set A in R^n. For x in A, what is the smallest
dimension d s.t. there is a C^1 manifold that contains all points in A that
lie in some neighborhood of x? We define a tangent space T_x(A) in a
natural way and show that the answer is d = dim (T_x(A)).