MD-PS 2002 Talks Titles and Abstacts

# TITLES AND ABSTRACTS OF TALKS

## Maryland-Penn State Semiannual Workshop 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.
In this talk we shall discuss these and other similar results from the perspective of Ergodic Ramsey Theory. This discussion will lead us to new interesting results and conjectures. In particular we will see the foregoing as a special case of the appearance of rather arbitrary finite configurations inside sufficiently large sets.

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.

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