Maryland-Penn State Semiannual Workshop
on Dynamical Systems and Related Topics

University of Maryland, March 20-23, 2004

There are links to pdf files of the talks of S. Adams, M. Einsiedler and B. Kra, following their abstracts below.

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
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.

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