PSU Mark
Eberly College of Science Mathematics Department

Abstracts

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DYNAMICAL SYSTEMS and RELATED TOPICS WORKSHOP
October 14-17, 2004
Penn State


ABSTRACTS OF TALKS

 

 

 

Mike Boyle (University of Maryland)

Almost isomorphism for countable state Markov shifts

Abstract. Among the countable state Markov shifts, the SPR (strongly positive recurrent, or stable positive recurrent) shifts are the most important distinguished class. For example, this is the class for which there is a measure of maximal entropy which is exponentially recurrent. Buzzi, Gomez and I have introduced a new notion (almost isomorphism) and shown that the SPR shifts are classified up to almost isomorphism by entropy and period. This has several applications. For example, entropy and period classify SPR shifts (as measurable systems with respect to the measure of maximal entropy) up to finitary isomorphism with finite expected coding time. Also, on account of earlier reductions by Buzzi to the SPR case, several classes of smooth or piecewise smooth systems are classified up to entropy conjugacy.

 

Vitaly Bergelson (Ohio State University)

Generalized polynomials: from Weyl and van der Corput to dynamical systems on nilmanifolds

Abstract. Generalized polynomials form a natural family of functions which are obtained from the conventional polynomials by the use of the greatest integer function, addition and multiplication. We will review some recent results on Diophantine approximations which extend the classical results of Weyl and van der Corput and discuss the connections between generalized polynomials and flows on nilmanifolds.

 

Jerome Buzzi (Ecole Polytechnique)

Quasi-finite puzzles

Abstract. Interval maps have a nice and natural symbolic dynamics. They are natural in that they are defined by their partitions into maximum interval of monotonicity. They are nice in that they are very close to subshifts of finite type. We try and generalize this to higher dimensional maps of the type:

$\displaystyle (x,y) \to (1.8-x^2+0.1\sin(2\pi y),1.5-y^2+0.1\sin(2\pi x)) $

 

by the way of dynamical puzzles a la Yoccoz. We show that these puzzles are close to subshifts of finite type in a precise sense, which in particular implies estimates on their associated zeta functions.

 

Chris Connell (Indiana University)

Prescribing boundaries for random walks on hyperbolic groups

Abstract. For a random walk on a group, Furstenberg associated to it a Poisson boundary; a measure theoretic representation space for bounded harmonic functions. For a Gromov hyperbolic group $ G$ with geodesic boundary $ B$ , we consider the inverse problem of starting with a measure $ m$ on $ B$ and asking whether or not $ (B,m)$ can arise as a Poisson boundary for some random walk on $ G$ . For $ G$ a CAT(-1) group and m any Lipschitz quasiconformal measure, or more generally a measure from a large family of classes of conditional Gibbs states, we give an affirmative answer. For general groups acting on Gromov hyperbolic spaces we show that continuous quasiconformal measures on the ``radial" limit set arise as $ G$ -quotients of the Poisson boundary. The proofs also establish approximation results generalizing those of Bonsall and Hayman-Lyons. This is joint work with Roman Muchnik.

 

Dmitry Dolgopyat (University of Maryland)

Regularity of transport coefficients

Abstract. We review results about smoothness of dependence of dynamical invariants on parameters and present some applications and open problems.

 

Andrew Dykstra (University of Maryland)

Good almost conjugacy for G-shifts of finite type

Abstract. Let $ G$ be a finite group. A $ G$ -shift of finite type ($ G$ -SFT) is an SFT equipped with a continuous shift-commuting G action. Answering a question of Parry, we classify irreducible G-SFTs up to right-closing almost conjugacy (RCAC). In particular, for mixing G-shifts of finite type where the $ G$ action is free, TFAE:

  1. entropy and ideal class agree,
  2. the $ G$ -SFTs are right closing almost conjugate as SFTs,
  3. the $ G$ -SFTs are right closing almost conjugate.

In the general irreducible case, period and one additional invariant are also needed for the classification. The equivalence relation RCAC is of interest because of its connects with algebraic invariants, resolving maps and the measurable relation of regular isomorphism. This talk is based on ``Right closing almost conjugacy for G-shifts of finite type"; a preprint is available on my webpage (Google: Andrew Dykstra).

 

Patrick Eberlein (University of North Carolina)

Geometric and dynamical properties of 2-step nilpotent Lie groups

Abstract. Let $ N$ be a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric, and let $ \mathfrak{N}$ denote its Lie algebra. We say that $ N$ has type $ (p,q)$ if the commutator ideal of $ \mathfrak{N}$ has dimension p and codimension $ q$ . Let $ \mathfrak{s}\mathfrak{o}$ (q, $ \mathbb{R}$ ) denote the $ q \times q$ skew symmetric matrices with real entries. If $ N$ has type $ (p,q)$ , then $ \mathfrak{N}$ is isomorphic to $ \mathfrak{N}' = \mathbb{R}^q \oplus W$ (direct sum), where $ W$ is a p-dimensional subspace of $ \mathfrak{s}\mathfrak{o}$ (q, $ \mathbb{R}$ ) and $ \mathfrak{N}$ ' has a simple bracket relation for which $ [\mathbb{R}^q, \mathbb{R}^q]=W$ and $ W$ lies in the center of $ \mathfrak{N}$ . We begin by describing some left invariant first integrals for the geodesic flow on the unit tangent bundle SN. This is accomplished by using the canonical Poisson structure on $ \mathfrak{N}$ and finding first integrals of the energy Hamiltonian vector field on $ \mathfrak{N}$ . Linear and quadratic first integrals on $ \mathfrak{N}$ are classified. One of the most interesting quadratic examples includes the Ricci tensor of N, regarded as a symmetric, bilinear form on $ \mathfrak{N}$ . Let $ \{C^1 ,\dots , C^p\}$ be the $ q \times q$ skew symmetric structure matrices that arise from a basis of $ \mathfrak{N}' = \mathbb{R}^q \oplus W$ that is a union of bases from $ \mathbb{R}^q$ and $ W$ . Let $ V = \mathfrak{s}\mathfrak{o}(q,\mathbb{R}) \times\cdots \times \mathfrak{s}\mathfrak{o}(q,\mathbb{R})$ ($ p$ times), and let $ C$ be the element $ (C^1, \dots , C^p)$ in $ V$ . The group $ GL(q,\mathbb{R}) \times GL(p,\mathbb{R})$ acts in a natural way on $ V$ . We consider the existence of left invariant metrics $ \langle , \rangle$ on $ N$ with two types of special Ricci tensor, one of which is the property of being a first integral for the geodesic flow. Metrics $ \langle , \rangle$ with these two types of Ricci tensor exist if and only if the $ H$ orbits of $ C$ in $ V$ are closed, where $ H = SL(q,\mathbb{R}) \times SL(p,\mathbb{R})$ and $ SL(q,\mathbb{R})$ respectively. We describe a criterion due to R. Richardson and P. Slodowy for an H orbit in V to be closed, where H is a self adjoint subgroup of $ GL(q,\mathbb{R}) \times GL(p,\mathbb{R})$ . $ N$ admits a lattice $ L$ if and only if $ \mathfrak{N}$ admits a basis $ \mathfrak{B}$ whose structure constants are integers ( $ \mathbb{Z}$ -structure). The problem of finding Anosov diffeomorphisms on $ L \backslash N$ reduces to the algebraic problem of finding automorphisms h of $ \mathfrak{N}$ such that h leaves invariant $ \mathbb{Z}$ -span( $ \mathfrak{B}$ ) and $ \det(h) = 1$ or $ -1$ . We describe an existence result for Anosov diffeomorphisms due to J. Lauret, and we also describe some nonexistence results. In particular, the nonexistence results hold for generic examples of type (p,q), where $ \mathfrak{N}' = \mathbb{R}^q \oplus W$ , unless the pair $ (p,q)$ belongs to a small explicit list. Nonexistence also holds when $ W$ is of Clifford type or when $ W = \mathfrak{s}\mathfrak{o}(3,\mathbb{R})$ and $ q$ is even. Rational structures on $ \mathfrak{N}$ correspond bijectively to commensurability classes of lattices in $ N$ . Two rational structures on $ \mathfrak{N}$ are said to be equivalent if they differ by an automorphism of $ \mathfrak{N}$ . If $ \mathfrak{N}$ of type $ (p,q)$ has a rational structure, then $ \mathfrak{N}$ is isomorphic to $ \mathbb{R}^q \oplus W$ , where $ W$ has a basis of elements with rational entries. Moreover, $ \mathbb{R}^q \oplus W$ admits a basis as discussed above where the entries of the structure element $ C = (C^1, \dots , C^p)$ in $ V = \mathfrak{s}\mathfrak{o}(q,\mathbb{R}) \times\cdots \times \mathfrak{s}\mathfrak{o}(q,\mathbb{R})$ ($ p$ times) are rational. Let $ G$ denote $ GL(p,\mathbb{R}) \times GL(q,\mathbb{R})$ and let $ G_{Q}$ denote $ GL(p,\mathbb{Q}) \times GL(q,\mathbb{Q})$ . The space of rational structures on $ \mathfrak{N}$ can be identified with $ V(Q,C) / G_{Q}$ , where $ V(Q,C)$ consists of the elements with rational entries in the orbit $ G(C)$ in $ V$ .

 

Manfred Einsiedler (Princeton)

Rigidity of measures - the high entropy case and joinings

Abstract. In this talk we will discuss two separate joint projects with A. Katok and E. Lindenstrauss respectively.

For measure rigidity of the Cartan action on $ SL(3,\mathbb{R})/SL(3,\mathbb{Z})$ two separately developed techniques have been used: the high and low entropy argument. In a recent joint work with A. Katok we have generalized the former to non-split groups where we also include locally homgeneous spaces defined by products of real and $ p$ -adic Lie groups. Moreover, we do not assume that our action is defined by the full Cartan subgroup. This is important in the application to joinings of Cartan actions which is a joint work with E. Lindenstrauss: any ergodic joining between two Cartan actions on $ SL(3,\mathbb{R})/\Gamma$ is algebraic.

 

Renato Feres (Washington University)

Random walks derived from billiards

Abstract. We introduce a class of random dynamical systems derived from billiard maps, which we call ``random billiards," and study certain random walks on the line obtained from them. The interplay between the billiard geometry and the stochastic properties of the random billiard is investigated. Our main results are concerned with the description of the spectrum of the random billiard's Markov operator and the characteristics of diffusion limits under appropriate scaling.

 

Angela Grant (University of Maryland)

Finding optimal orbits of chaotic systems

Abstract. Chaotic dynamical systems can exhibit a wide variety of motions, including periodic orbits of arbitrarily large period. We consider the question of which motion is optimal, in the sense that it maximizes the average over time of some given scalar ``performance function". Past work has shown that optimal motions tend to be periodic orbits with low period but does not describe, beyond a brute force approach, how to determine which orbit is optimal in a particular scenario. For one-dimensional expanding maps, we have developed a constructive method for computing the optimal average and corresponding periodic orbit. We demonstrate that this method works quite well in practice and discuss progress toward a method for higher dimensional systems.

 

Sergei Ivanov (Russian Academy of Sciences)

Boundary rigidity and volume minimality for almost flat metrics

Abstract. A compact Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined (up to an isometry) by the distances between the boundary points; and is said to be a minimal filling if it has the least volume among all compact Riemannian manifolds with the same boundary and the same or greater boundary distances. I will discuss the following result of a recent joint work with D.Burago: Euclidean regions with Riemannian metrics sufficiently close to a Euclidean one are minimal fillings and boundary rigid. The proof is based on Gromov's idea of representing a Riemannian manifold as a surface in an $ L$ -infinity Banach space, and an observation that the resulting surfaces resemble (in some sense) minimal surfaces in Euclidean spaces.

 

Dmitry Kleinbock (Brandeis University)

Measure rigidity and p-adic Littlewood-type problems

Abstract. I will explain how recent work of Einsiedler-Katok-Lindenstrauss gives rise to new results and conjectures in multiplicative Diophantine approximation over the product of real and $ p$ -adic fields. Joint with Manfred Einsiedler.

 

Bruce Kleiner (University of Michigan/NYU)

Mostow rigidity for hyperbolic groups

Abstract. I will discuss a rigidity theorem which generalizes Mostow rigidity to a class of Gromov hyperbolic groups. This may viewed as a first step toward attaching a canonical geometry to (certain) hyperbolic groups. The setting, theorems, and proofs involve recently developed techniques from analysis on metric spaces.

 

Boris Kruglikov (University of Tromsoe)

Strictly non-proportional geodesically equivalent metrics have $ h_$top$ (g)=0$

Abstract. This is a joint result with Vladimir Matveev.

Let $ g$ be a Riemannian metric on a closed manifold $ M$ and $ f:M\to M$ be a projective transformation, i.e. a diffeomorphism preserving non-parametrized geodesics. Assume that it is non-degenerate in a sense that the endomorphisms field $ g^{-1}f^*g$ of $ TM$ has simple spectrum at some point of $ M$ (i.e. metrics $ g$ and $ f^*g$ are strictly-non-proportional at this point). Then the topological entropy of the geodesic flow of $ g$ vanishes.

As a corollary we conclude that a rationally hyperbolic closed manifold does not admit two geodesically equivalent Riemannian metrics, which are strictly non-proportional somewhere. Morover, in the non-simply connected case a manifold admitting such a pair of metrics can be covered by the product of a rationally-elliptic manifold and a torus.

The obtained results are sharp: there are geodesic flows with $ h_$top$ (g)>0$ admitting (degenerate in the above sense) projective transformations and many examples of rationally elliptic manifolds with non-trivially geodesically equivalent metrics are known.

 

Francois Ledrappier (University of Notre Dame)

Martin points at infinity for rank one manifolds

Abstract. The Martin boundary of a Riemannian domain is a very precise way of compactifying the domain using potential theory of the Laplacian. So precise that not much is known. I'll try to give some background, and I'll present a Work in Progress partial result with Jianguo Cao and Huijun Fan about the case of a simply connected nonpositively curved manifold having a cocompact group of isometries and at least one isometry with a hyperbolic axis (Ballmann rank one manifolds).

 

John Milnor i(SUNY Stony Brook)

Rational maps of $ P^2$ , examples and problems

Abstract. This talk will describe joint work with Araceli Bonifant and the late Marius Dabija, who introduced the study of rational maps of $ P^2$ with an invariant elliptic curve. This is an extremely rich collection of examples, with widely varying dynamic behavior. I will focus on explicit examples and problems.

 

William Minicozzi (Johns Hopkins University)

The Calabi-Yau conjectures for embedded surfaces

Abstract. I will discuss the proof of the Calabi-Yau conjectures for embedded minimal surfaces. This is joint work with Toby Colding.

The original form of these conjectures was given in 1965 by E. Calabi:

``Prove that a complete minimal hypersurface in $ \mathbb{R}^n$ must be unbounded.''

and

``A complete minimal hypersurface in $ \mathbb{R}^n$ has an unbounded projection in every $ (n-2)$ -dimensional flat subspace.''

The immersed versions of these conjectures turned out to be false; immersed counterexamples were constructed by Jorge and Xavier in 1980 and Nadirashvili in 1996. We will show that the embedded versions are true.

Our main result is an effective version of properness for disks, giving a chord arc bound. Obviously, intrinsic distances are larger than extrinsic distances, so the significance of a chord arc bound is the reverse inequality, i.e., a bound on intrinsic distances from above by extrinsic distances. This chord-arc bound immediately implies both of Calabi's conjectures for embedded minimal disks.

 

Xiaochun Rong (Rutgers University)

The semi-rigidity of nonpositively curved metrics

Abstract. We call the moduli space of non-positively curved metrics on a closed manifold rigid if all the metrics are isometric up to a rescaling. The moduli space is rigid if it contains an irreducible metric of rank at least two (the higher rank rigidity).

We call a moduli space semi-rigid, if all metrics are `alike' in the following sense: each metric poses a certain compatible local splitting structure whose underlying topological structure is independent of the metric. A typical example is the moduli space of nonpositively curved metrics on a graph $ 3$ -manifold. The semi-rigidity is closely related to the collapsing theory of Cheeger-Gromov. In this talk, we will discuss the semi-rigidity and its applications.

 

Nandor Simanyi (University of Alabama, Birmingham)

The Boltzmann-Sinai Ergodic Hypothesis in two dimensions (without exceptional models)

Abstract. We consider the system of $ N$ ($ \ge2$ ) elastically colliding hard balls of masses $ m_1,\dots,m_N$ and radius $ r$ in the flat unit torus $ {\mathbb{T}}^\nu$ , $ \nu\ge2$ . In the case $ \nu=2$ we prove (the full hyperbolicity and) the ergodicity of such systems for every selection $ (m_1,\dots,m_N;r)$ of the external geometric parameters, without exceptional values. In higher dimensions, for hard ball systems in $ {\mathbb{T}}^\nu$ ($ \nu\ge3$ ), we prove that every such system (is fully hyperbolic and) has open ergodic components.

 

Andrew Török (University of Houston)

Stability of rapid mixing (joint with M. Field and I. Melbourne)

Abstract. We obtain general results on the stability of rapid (superpolynomial) decay of correlations for hyperbolic flows: amongst the $ C^r$ Axiom A flows, there is a $ C^2$ -open, $ C^r$ -dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing.

For nontrivial attracting hyperbolic basic sets, we obtain a $ C^1$ -open, $ C^r$ -dense set of rapid mixing flows.

These are consequences of a technique used to prove stable ergodicity of group extensions, and the characterization of rapid mixing flows by Dolgopyat.

 

Anatoly Vershik (Russian Academy of Sciences)

Universal metric space, random graph and related questions

Abstract. We will describe the classical results of Urysohn about the universal metric space, and Rado-Erdos-Renyi about the universal or random graph. The definition of random metric spaces will be given, and it will be explained why with probability one the random metric space is universal.

 

Anatoly Vershik (Russian Academy of Sciences)

Metric spaces with measures and classification of function of several variables

Abstract. We give a classification of metric spaces with measures and measurable functions of several variables. We connect this question with the problem about invariant distributions of the random matrices.

 

Todd Young (University of Ohio)

A geometric proof of separatrix crossing results

Abstract. We investigate the adiabatic invariance of the action variable of one degree of freedom time-dependent Hamiltonian systems when trajectories cross a separatrix and give relatively simple, geometrically motivated proofs. This is joint work with Shui-Nee Chow.

 

Hong-Kun Zhang (University of Alabama, Birmingham)

Billiards with polynomial mixing rates (joint with N. Chernov)

Abstract. While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic - enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. However, mathematical methods for the analysis of systems with slow mixing rates were developed just recently and are still difficult to apply to realistic models. Here we reduce those methods to a practical scheme that allows us to obtain a nearly optimal bound on mixing rates. We demonstrate how the method works by applying it to several classes of chaotic billiards with slow mixing as well as discuss a few examples where the method, in its present form, fails.

 

Anatole Katok (Penn State)

Measure rigidity beyond uniform hyperbolicity and positive entropy

Abstract. In the last decade, and especially in the last few years a great progress has been achieved in the classification of invariant ergodic measures of algebraic Anosov and to a lesser extent partially hyperbolic actions of higher rank abelian groups, i.e. $ \mathbb{Z}^k$ and $ \mathbb{R}^k$ for $ k\ge 2$ . At the same time progress in the differentiable rigidity for hyperbolic actions of such groups indicates that extension of the measure rigidity theory in this direction may turn out to be essentially vacuous and may not be worth pursuing. On the other hand, there are actions of higher rank abelian groups with hyperbolic invariant measures, i.e. measures with non-vanishing Lyapunov exponents. Topological structure of such actions may be rather complicated but some rigidity appears for hyperbolic invariant measures with positive entropy. An essential modification of the methods of the measure rigidity theory is needed to tackle this case. I will discuss the results for measures for Cartan type (actions of $ \mathbb{Z}^{n-1}$ on $ n$ -dimensional manifolds) with Lyapunov exponents in general position which I have obtained several years ago but whose detailed proofs are still not published. I will outline open problems and possible approaches to more general situations.

Another notoriously resistant problem concerns zero entropy measures even for such simple systems as the Furstenberg $ \times2,\,\,\times 3$ action on the circle. I offer some speculations on how partial progress in this direction might be achieved.

 

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