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

University of Maryland, March 19-22, 2005

Jose Alves,   University of Porto
Smooth conjugacies of interval maps
      We consider a topological conjugacy between interval maps. We show that for certain maps if the conjugacy is differentiable at some point, then it is differentiable in an open interval.

Sarah Bailey ,   University of North Carolina at Chapel Hill
PhD student, advised by Karl Petersen
Computing dimension groups for a certain family of non-simple Bratteli diagrams
      Dimension groups of simple Bratteli diagrams are known to be order isomorphic to the continuous functions modulo the coboundaries. I will discuss a special family of non-simple Bratteli diagrams for which the dimension groups are computable. In this family of Bratteli diagrams the adic transformations are not continuous, but the dimension groups are still order isomorphic to the continuous functions modulo the continuous coboundaries.

Youngna Choi, Montclair State University ,  
Topology of Attractors from Two-Piece Expanding Maps
      In this paper we study the topology of the invariant set L_f derived from an expanding map with one discontinuity f. We classify the periodic orbits on the boundary of L_f that enable L_f to have a trapping region and that do not. We also show that when there are two eventually periodic orbits on the boundary of L_f, the components of L_f form a Markov partition, and form its transition matrix we can calculate the precise entropy of the map. Furthermore, we show that the set of attractors derived from two piece expanding maps is open and dense in the set of all invariant sets derived by the maps.

Alex Clark ,   University of North Texas
Coding expansive automorphisms of groups and Rauzy fractals
      Reporting on joint work with Robbert Fokkink, we will present recent results on the coding of expansive automorphisms of compact groups. Making use of the (non)-archimedean evaluations at the roots of an associated polynomial, we will draw connections with generalized Rauzy fractals.

Manfred Einsiedler   Universitaet Wien, Princeton University
Measure Rigidity for Cartan actions
      In an ongoing joint work with Elon Lindenstrauss we describe invariant measures with positive entropy for the maximal split torus on a locally homogeneous space defined by a reductive linear algebraic group (also known as Cartan actions or Weyl chamber flows). This generalizes earlier work of Katok, Lindenstrauss and myself regarding SL(n,R)/Gamma to not necessarily split and not necessarily real groups. Moreover, if the measure is not algebraic we give restrictions on the support of the measure.

Nikos Frantzikinakis,   Penn State University
Sets of multiple recurrence and convergence
      For every positive integer k we will construct a set of k-recurrence but not (k+1)-recurrence. This extends a result of Furstenberg who produced a set of 1-recurrence but not 2-recurrence. We will also discuss a similar result for convergence of multiple ergodic averages and point out a combinatorial consequence related to Szemeredi's theorem on arithmetic progressions.
      This is joint work with E. Lesigne and M. Wierdl.

Daniel Genin ,   Penn State University
PhD student, advised by Serge Tabachnikov
Hyperbolic outer billiards
      Mathematical billiards are an extensively studied and in many respects well understood dynamical systems. For instance, the question of chaotic behavior was very well explored since Sinai described the first class of hyperbolic billiards in 1970. Outer billiards are a much less well known and understood class of dynamical systems with interesting connections to ordinary billiards and impact oscillators. After a brief survey of chaotic ordinary billiards I will describe the first example of a chaotic outer billiard.

Huyi Hu ,   Michigan State University
Convergence rates of the transfer operators for sigma finite measures
      Consider piecewise expanding maps on the unit interval with indifferent fixed point. It is well known that the convergence rates to equilibruim under the transfer operator is polynomial if the systems admit absolutely continuous probability measure. We study the case that the invariant measure is sigma finite, and give both upper and lower bound estimates for the convergence rates.
      This is a joint work with Nicolai Haydn.

Anatole Katok ,   Penn State University
New geometric method in differentiable rigidity of partially hyperbolic actions
      I will explain the main ingredients of the proof of local differentiable rigidity (up to a finite-dimensional family of standard algebraic perturbations) for restrictions of the Weyl chamber flow on SL(n,R) to a plane in general position. The method of proof is radically different and complementary to the KAM iteration scheme which has been successfully applied to actions by partially hyperbolic automorphisms of the torus. It uses, among other things, ideas and results from algebraic K-theory. This is a part of an ongoing joint work with Danijela Damjanovic.

Elon Lindenstrauss ,   Princeton University
On measures invariant under tori and other actions
      (Math department colloquium, Friday March 18)
      An important open problem in the theory of flows on locally homogeneous spaces is the classification of measures invariant under actions of multidimensional diagonalizable groups. A similar classification due to Ratner for measures invariant under actions of unipotent groups has had numerous applications in many areas, and already the partial results we have towards measure classification for diagonalizable actions have found some applications in number theory and arithmetic quantum chaos.
      I will survey some recent results regarding these invariant measures and their applications.

Hee Oh ,   Cal Tech
Equidistribution of solvable flows
      We will explain the equidistribution result of certain solvable groups on $G/\Gamma$ (as usual, $G$ is semisimple and $\Gamma$ a lattice). More precisely, if $H$ is a semisimple subgroup of $G$ with $H=KAN$ an Iwasawa decomposition, and the orbit $Hx$ is dense in $G/\Gamma$, then $Qx$ is equidistributed in $G/\Gamma$ for any closed subgroup $Q$ of $H$ containing $AN$. As an application, we obtain the equidistribution of lattice orbits on the Furstenberg boundary of $G$. Moreover we show that this equidistribution on the Furstenberg boundary is independent from the distribution of lattice orbits in the symmetric space. This is a joint work with Alex Gorodnik.

Jens Marklof ,   Bristol
Limit theorems for skew translations
Strongly chaotic systems such as Anosov maps are well known to satisfy central theorems for the distribution of long trajectories. The case of ergodic, but not mixing, skew translations seems more subtle, and I will discuss some of the recent developments and open conjectures.

Kyewon Park ,   Ajou University
Complexity of entropy zero dynamical systems
      Motivated by the study of actions of Z^2 and more general groups, and their subgroup actions, we investigate entropy type invariants for deterministic systems. Several refinements of the notion of entropy have been introduced by various authors. We propose a new notion, the entropy dimension and develop the basic definitions. We consider their behaviors on examples in the case of Z actions and Z^2 actions.

Karl Petersen ,   University of North Carolina at Chapel Hill
The adic transformation on the Euler graph
      The Euler graph has vertices labeled (n,k) for n=0,1,2,... and k=0,1,...,n, with k+1 edges from (n,k) to (n+1,k) and n-k+1 edges from (n,k) to (n+1,k+1). The number of paths from (0,0) to (n,k) is the Eulerian number A(n,k), which is the number of permutations of 1,2,...,n+1 with exactly n-k falls and k rises. The symmetric measure is defined by assigning equal weights 1/(n+2) to all of the edges downward from each level n. We prove that the adic transformation on the Euler graph is ergodic with respect to the symmetric measure. The argument adapts a previous proof by Keane of ergodicity of the symmetric Bernoulli measure for the Pascal adic by analyzing the relevant random walk. A study of the asymptotic properties of the Eulerian numbers yields the stronger result that in fact the symmetric measure is the unique fully supported ergodic invariant Borel probability measure for this system. These results are part of an ongoing joint project with Sarah Bailey, Michael Keane, and Ibrahim Salama in various combinations.

Mark Pollicott,   University of Warwick, England
Dynamical zeta functions and Dilation Equations
      Dynamical zeta functions (in the sense of Ruelle) often provide a very useful tool for studying characteristics associated to hyperbolic dynamical systems (e.g., Lyapunov exponents, Hausdorff Dimension, etc.). In particular, this can often lead to an effective method for their computation. We want to apply this approach to the study of the regularity of solutions to Dilation Equations f(x) = \sum_k c_k f(2x - k) on the real line, developing an idea of Cohen and Dauberchies. This has applications to the theory of Wavelets, and is joint work with Howie Weiss.

Mrinal Roychowdhury,   Wesleyan University
PhD student, advised by Mike Keane
Finitary orbit equivalence
      Two invertible dynamical systems $(X, \mathfrak{A}, \mu, T)$ and $(Y, \mathfrak{B}, \nu, S)$ are said to be orbit equivalent if there exists an invertible measure-preserving mapping $\phi$ from a subset of $X$ of measure one to a subset of $Y$ of full measure such that $\phi(\text{Orb}_T(x))=\text{Orb}_S\phi(x)$ for $\mu$-almost every $x\in X$. Suppose that the above $X$ and $Y$ are also topological spaces and that $T$ and $S$ are homeomorphisms. We say that the orbit equivalence is finitary if both $\phi$ and $\phi^{-1}$ are continuous after restriction to sets of measure 1. I will discuss how an irrational rotation and the binary odometer are finitarily orbit equivalent.

Daniel J. Rudolph ,   Colorado State University
Mixing, disjoint factors and isometric extensions
      Recently Francois Parreau and Jean-Paul Thouvenot have proven the rather startling fact that any ergodic transformation that is not mixing has a nontrivial factor disjoint from all mixing actions. This result has a relative version and with it one is able to prove a relative version of the fact that weakly mixing isometric extensions of mixing actions must be mixing. This can then be used to show that for discrete amenable group actions, weakly mixing isometric extensions of mixing actions must be mixing, using the orbit transference method.
      My goal is to sketch the machinery of the Parreau-Thouvenot argument and how it can used to carry out this program.

Omri Sarig,   Penn State University
Invariant Radon measures for the horocycle flow on periodic surfaces
      This is joint work with Francois Ledrappier.
      The horocycle flow of a non-geometrically finite hyperbolic surface may have many different (infinite) ergodic invariant Radon measures. (This should be contrasted with co-compact or co-finite volume case where there is just one non-trivial measure - up to scaling).
      We classify these measures for the class of periodic surfaces: regular covers of surfaces of finite volume. It turns out that in this case there are as many measures as there are positive eigenfunctions for the Laplacian of the surface.

Nandor Simanyi ,   University of Alabama at Birmingham
Rotation sets of billiards
      So far we have studied the rotation sets of two families of billiards:
      (A) The inertia motion of a point in the flat torus $\Bbb T^m$ minus a strictly convex, compact obstacle $O$ with a smooth boundary. Here, by definition, the rotation set $R$ consists of all limiting points of the average displacements along orbit segments, when the lengths of these segments tend to infinity.
      (B) The inertia motion of a point in a rectangle minus a strictly convex, compact obstacle $O$ with a smooth boundary inside the interior of the rectangle. Here the rotation set $R$ is, by definition, the set of all limiting values of the average wrapping around the obstacle by orbit segments whose lengths tend to infinity.
      We will present a detailed description of the above rotation sets. The proofs use ideas from geometry, topology, and a bit of combinatorics.
      (These are joint results with A. Blokh and M. Misiurewicz.)

Howard Weiss,   Penn State University
How smooth is your wavelet: a thermodynamic formalism approach
      The study of wavelet regularity is currently a major topic of investigation having important applications. In this talk I will discuss our unified approach to wavelet regularity via thermodynamic formalism. The first part of the talk will be a quick introduction to wavelets.
      This is joint work with Mark Pollicott.

Tom Ward ,   University of East Anglia
Disjointness and entropy geometry
      This is joint work with Manfred Einsiedler, in which the entropy geometry of commuting automorphisms of zero-dimensional groups is used to show that many natural actions are mutually disjoint.

Tamar Ziegler ,   Ohio State University
Configurations in sets of positive upper density in R^m
      We apply ergodic theoretic tools to solve a classical problem in geometric Ramsey theory: let E be a measurable subset of R^m, with positive upper density. Let V={0,v_1,...,v_k} be a subset of R^m. We show that for r large enough, we can find an isometric copy of rV arbitrarily close to E. This is a generalization of the work of Furstenberg, Katznelson and Weiss proving this property for k=m=2.

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