PSU Mark
Eberly College of Science Mathematics Department

Abstracts

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DYNAMICAL SYSTEMS and RELATED TOPICS WORKSHOP DYNAMICS AT THE CROSSROADS OF MODERN MATHEMATICS: RECENT PROGRESS AND PERSPECTIVES.

October 2-5, 2006
Penn State


ABSTRACTS OF TALKS




Boris Begun (Hebrew University)

Bernoulli-type behavior in random dynamical systems.

Abstract. This is a joint work with A. del Junco. Consider a measure-preserving transformation of a probability space. A finite partition of the space is called weakly independent if there are infinitely many images of this partition under powers of the transformation that are jointly independent. Krengel proved that a transformation is weakly mixing if and only if weakly independent partitions of the underlying space are dense among all finite partitions. Using the tools developed in the previous papers of del Junco - Reinhold - Weiss and del Junco - Begun we obtain Krengel-type result for weakly mixing random dynamical systems (or equivalently, skew products that are relatively weakly mixing).


Yuri Burago (St. Petersburg)

Bi-Lipschitz equivalence of Alexandrov surfaces and global Chebyshev coordinates.

Abstract. We study the following problem: under which conditions Riemannian 2-manifolds (or, more generally, Alexandrov surfaces) are bi-Lipschitz equivalent with a controlled Lipschitz constant? Such conditions include bounds on diameter, systolic constant, total curvature and its distribution over the manifolds. We also prove existence of global Chebyshev coordinate on a (complete simply connected) Alexandrov surface under optimal curvature bounds. Such coordinates are useful for constructing bi-Lipschitz maps between surfaces.


Mark Demers (Fairfield University)

Stability of statistical properties for piecewise hyperbolic maps.

Abstract. I will discuss the statistical properties of piecewise smooth dynamical systems by studying directly the action of the transfer operator on appropriate spaces of distributions. In the case of two-dimensional maps with uniformly bounded second derivative, we obtain a complete description of the SRB measures, their statistical properties and their stability with respect to a large class of perturbations, including deterministic and random perturbations, as well as holes. This is based on joint work with C. Liverani.


Dmitry Dolgopyat (Penn State and Maryland)

Recurrence properties of Lorenz process.

Abstract. Lorentz process is one of the simplest Hamiltonian system with infinite measure. It is well known that ergodic properties of infinite measure preserving transformation are closely related to recurrence properties of the system. In this talk we discuss the recurrence proprties of Lorentz process and present several applications to systems which locally look like a Lorentz process.


Todd Fisher (Maryland)

Superexponential growth of the number of periodic orbits for non-hyperbolic homoclinic classes.

Abstract. We show there is a residual subset $ S(M)$ of diffeomorphisms of $ M$ such that, for every diffeomorphism in $ S(M)$,   any homoclinic class containing periodic saddles of different indices has superexponential growth of the number of periodic points inside the homoclinic class. This is joint work with Christian Bonatti and Lorenzo Diaz.


Hillel Furstenberg (Hebrew University)

$ (G,\mu)$-dynamical systems: structure theory and multiple recurrence.

Abstract. A $ (G,\mu)$-dynamical system is a pair $ (X,\nu)$ where $ X$ is a $ G$-space and $ \nu$ is a "stationary" probability measure on $ X$ for the measure $ \mu$ on $ G$: $ \mu*\nu = \nu$.   We discuss some of the contexts in which these appear and pose some general questions regarding these systems. As an example we discuss a structure theorem valid for all such systems and present an application to multiple recurrence for any minimal action of $ G$ where $ G$ is the group $ Sl_2(\mathbb{R})$.


Anish Ghosh (University of Texas)

$ SL(2,k)$-ergodic invariant measures in positive characteristic.

Abstract. This is joint work with M.Einsiedler. Under some mild restrictions on characteristic, we prove a special case of Ratner's measure classification theorem for arbitrary local fields. This answers in part a question raised by M.Ratner.


Rostislav Grigorchuk (Texas A& M)

Amenability, Liouville property and ergodicity of boundary actions.

Abstract. We will start with cogrowth criterion of amenability and explain how it was used to treat the alternative amenable/nonamenable in the case of discrete groups and how the first nontrivial estimations of Tarski number were obtained. Then the self-similar groups will come to the arena and we will explain why they and their Schreier graphs are useful in study of amenability and random walks. Then we will switch to topics around Poisson Boundary and Liouvulle property for groups and graphs and study the ergodic properties of boundary actions of a free group and of its subgroups. The Nielsen and Schreier-Redeimeister methods as well as cogrowth will be used for description of partitioning into conservative and dissipative parts.


Boris Kalinin (University of South Alabama)

Smooth rigidity of higher rank abelian actions.

Abstract. We will consider rigidity problems related to the smooth structure of hyperbolic actions of groups $ \mathbb{Z}^k$ and $ \mathbb{R}^k$ with $ k$ at least 2. We will discuss recent progress with emphasis on global smooth rigidity as well as the relations with other problems for higher rank actions and for single hyperbolic and partially hyperbolic systems.


Dmitry Kleinbock (Brandeis)

Expanding translates of horospheres and their applications to number theory.

Abstract. I will state a new result on equidistribution of expanding translates of orbits of horospherical subgroups on homogeneous spaces, and discuss two proofs, joint with Barak Weiss and Gregory Margulis, respectively, and, if time allows, two applications to the theory of "diophantine approximation with weights".


Bruce Kleiner (Yale)

BiLipschitz embedding in Banach spaces.

Abstract. Motivated by a conjecture from computer science, I will discuss biLipschitz embeddings $ X \to V$ where $ X$ is a metric space and $ V$ is a Banach space. The focus will be on the case when $ X$ is a metric measure space satisfying a Poincare inequality. Of particular interest is the case when the target Banach space is $ L^1$,  in which case there is a new link between embedding questions and the structure of sets of finite perimeter in $ X$.   By exploiting recent work on geometric measure theory in the Heisenberg group, we show that the Heisenberg group cannot be biLipschitz embedded in $ L^1$,   confirming a conjecture of Assaf Naor.

This is joint work with Jeff Cheeger.


Gerhard Knieper (Ruhr University at Bochum)

Hyperbolic dynamics and Riemannian geometry.

Abstract. The purpose of this lecture is to give a survey on geodesic flows with some weak conditions on hyperbolicity. In the first part we will talk about the genericity of positive topological entropy for compact surfaces. In the second part we will focus on compact manifolds of nonpositive curvature. Their geodesic flows provide important examples of nonuniform - and partial hyperbolicity. In particular we will talk about the equidistribution of closed geodesics and the uniqueness of the measure of maximal entropy. Furthermore we will provide some applications to rigidity questions.


Svetlana Krat (Georgia Tech)

On surfaces with small variation of Gaussian curvature.

Abstract. Consider a surface with small variation of Gaussian curvature. One asks if this surface is actually a small perturbation of a developing surface (as opposed to the situation where there is no developing surface close to the original one). I have proven that a smooth compact surface in 3-dimensional Euclidean space with small variation of Gaussian curvature can be $ C^0$ (with respect to some parameterization) approximated by a smooth developing surface (with almost the same boundary).


Meera Mainkar (University of West Ontario)

Anosov automorphisms on nilmanifolds.

Abstract. Nilmanifolds admitting Anosov automorphisms play an important role in the theory of dynamical systems. These nilmanifolds correspond to Anosov Lie algebras of which few examples were known. We will discuss a combinatorial method using graphs of constructing Anosov Lie algebras. We will also describe some recent examples constructed using algebraic number theory.


Mark Pollicott (University of Warwick)

A dynamical analogue of Bauer's Theorem.

Abstract. Many years ago, we proved an analogue of the Chebotarov theorem in number theory, in the context of skew products over hyperbolic systems. We now consider a simple analogue of another such result, Bauer's theorem. This describes how the finite group in the extension is characterized in terms of which closed orbits split (i.e., lift to closed orbits of the same period). This was a collaboration with William Parry, FRS, who recently passed away.


Leonid Polterovich (Tel Aviv University)

Between symplectic dynamics and symplectic topology.

Abstract. Since symplectic revolution of the 1980-ies we witnessed a fruitful interaction between symplectic topology and symplectic dynamics. In the talk I shall illustrate some aspects of this interaction in a number of examples:

(1) Asymptotic invariants of symplectic maps: Their study uses the Floer theory, a powerful tool of modern symplectic topology.

(2) Function theory on symplectic manifolds: Nowadays this subject is making its first steps. Its surprising feature is appearance of seemingly elementary statements on functions which involve the uniform norm and the Poisson bracket. Their proofs are based on ideas and methods of symplectic dynamics.


David Ralston (Rice University)

An obesity epidemic in dynamical systems.

Abstract. The notion of heaviness involves comparing the expected value for a se- quence with the infimum of its averages. Though easy to define, this new tool raises some very interesting questions in ergodic theory and number theory. Basic existence theorems and some interesting initial investigations (in both dynamical systems and number theory) will be presented.


Roman Schubert (University of Bristol)

Mixing and equidistribution in wave propagation.

Abstract. We study solutions of the time dependent Schroedinger equation in the semiclassical limit $ \hbar\to 0$ with oscillating initial conditions. It is well known that the propagation of quantum states in the semiclassical limit is described by the underlying Hamiltonian dynamical system. We show that if this classical system is hyperbolic and mixing, then a certain class of quantum states become equidistributed for large times. The main problem in this field is to control the two limits $ \hbar\to 0$ and $ t\to \infty$ simultaneously.


Chistina Sormani (CUNY)

The topology of complete spaces with nonnegative Ricci curvature.

Abstract. While there have been many advances in the understanding of the topology of complete noncompact manifolds of nonnegative Ricci curvature, this area is wide open for further study. In particular, Milnor's famous 1969 conjecture that such a manifold has a finitely generated fundamental group is still open. The speaker will survey a number of theorems and examples, including her work with Zhongmin Shen classifying the codimension one integer homology of these manifolds and her proof of the Milnor Conjecture when the manifolds are assumed to have small linear diameter growth. Unlike the algebraic approach in Burkhard Wilking's reduction of the Milnor conjecture to manifolds with abelian fundamental groups and the analytic proof by Shing-Tung Yau that a manifold with positive Ricci curvature has trivial codimension one real homology, the proofs of these results are purely geometric and can be described with a few key diagrams and lemmas.


Corinna Ulcigrai (Princeton)

Mixing for flows over interval exchange transformations.

Abstract. We consider suspension flows over interval exchange transformations, under a roof function with logarithmic singularities. Such flows arise as minimal components of flows on surfaces given by multi-valued Hamiltonians. We prove that if the roof function has an asymmetric logarithmic singularity, the suspension flow is strongly mixing for a full measure set of interval exchanges. This generalizes results by Khanin and Sinai for flows over rotations of the circle and by Kochergin for flows with power-like singularities. In the proof we use a recent result by Avila-Gouzel-Yoccoz.


Jeremy Wang (University of Toronto)

An extension procedure for manifolds-with-boundary.

Abstract. Manifolds-with-boundary tend to be "negatively curved" objects since geodesics can bifurcate along the boundary. Nevertheless, there is a way to devise an extension procedure to locally isometrically embed any manifold-with-boundary into an Alexandrov space of curvature bounded below. This leads to upper estimates on volume, diameter, and dimension, as well as to precompactness and finiteness theorems. Then, time permitting, we will outline how this procedure, under certain curvature and injectivity radius bounds, can be applied to yield a fiber bundle structure for any manifold-with-boundary which is sufficiently Gromov-Hausdorff close to a given closed manifold.


Amie Wilkinson (Northwestern University)

Asymmetrical diffeomorphisms.

Abstract. Which diffeomorphisms of a compact manifold M commute with no other diffeomorphisms (except their own powers)? Smale asked if such highly asymmetrical diffeomorphisms are typical, in that they are dense in the Cr topology on the space of Cr diffeomorphisms Diffr(M). In this talk I will explain the recent (positive) solution to Smale's question for C1 symplectomorphisms and volume-preserving diffeomorphisms. I will also discuss progress on the general case. This is joint work with Christian Bonatti and Sylvain Crovisier.

Alistair Windsor (University of Texas, Austin)

Constructions in elliptic dynamics: recent results and open problems.

Abstract. We will discuss some recent results in the area of elliptic dynamical systems. This subject has considerable overlap with Diophantine approximation in number theory.

We will give some recent results for time changes of linear flows. The study of such time changes were originally proposed by Kolmogorov and form a prototype for KAM theory. We shall state some recent results and pose some open problems.

We will also discuss some recent results using conjugation by approximation methods and pose a number of open problems.

In both of these areas there is a dichotomy between the rigid phenomena associated with the Diophantine case and the plethora of examples obtained in the Liouville case.


Yuki Yayama (University of North Carolina)

On the uniqueness of measures of full Hausdorff dimension for some compact invariant sets

Abstract. The Hausdorff dimension of a "general Sierpinski cartpet" was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by considering a general Sierpinski carpet represented by a shift of finite type. Applying results of Ledrappier, Young and Shin, we study the Hausdorff dimension of a such a general Sierpinski carpet for the case when there is a saturated compensation function. We give some conditions under which a general Sierpinski carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measures.


Svetlana Katok 2006-10-23

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