PSU Mark
Eberly College of Science Mathematics Department

Abstracts

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DYNAMICAL SYSTEMS and RELATED TOPICS WORKSHOP

October 11-14, 2001
Penn State

ABSTRACTS OF TALKS



Michael Brin (University of Maryland)

On the integrabiblity of the central distribution of a partially hyperbolic diffeomorphism.

Abstract. We show that the central distribution of a partially hyperbolic diffeomorphism $f\colon M\to M$ is uniquely integrable if the stable and unstable foliations are quasi-isometric in the universal cover $\widetilde{M}$, i.e., if the distance in $\widetilde{M}$ between every two points, which lie in the same leaf, is bounded from below by a linear function of the distance along the leaf.


Richard Brown (American University)

The dynamics of surface diffeomorphisms on geometric structure.

Abstract. It is known that for S a compact surface, the modular group (i.e., the mapping class group when S is closed) of S acts symplectically on the symplectic leaves of the Poisson space of SU(2)-characters of representations of the fundamental group of S. Recently, it was established that this action is ergodic with respect to symplectic measure on these leaves. We discuss the dynamics of the actions of individual mapping classes on these character varieties, both explicitly for low genus surfaces, and implicitly for S a high genus surface.


Elizabeth Burslem (Northwestern University)

Centralizers of partially hyperbolic diffeomorphisms.

Abstract. Smale has conjectured that the generic diffeomorphism of a compact manifold has trivial centralizer - i.e. commutes only with powers of itself. The conjecture is known to be true for diffeomorphisms of a circle, and for structurally stable diffeomorphisms. We look at the case of partially hyperbolic diffeomorphisms - in particular perturbations of the time-t map of an Anosov flow, and partially hyperbolic skew products and their perturbations.


Leo Butler (Northwestern University)

Integrable hamiltonian systems with positive Lebesgue metric entropy.

Abstract. Examples are constructed of hamiltonian flows on Poisson manifolds which are integrable in the sense that there is an open, dense set of invariant tori on which the flow is conjugate to a translation-type flow; on the other hand, the metric entropy of the flow with respect to a canonical volume-form. It is shown that the set of invariant tori can be made arbitarily small.


Yitwah Cheung (Northwestern University)

Hausdorff dimension of the set of divergent trajectories in a product space.

Abstract. Let $g_t=diag(\exp(t),\exp(-t))$ act by right multiplication in each factor of the product space $X=(H\backslash G)^n$ where $G=SL_2({\mathbb R})$ and $H=SL_2({\mathbb Z})$. Let D be the subset of X consisting of those points whose orbit under the action leaves every compact set. We show that the Hausdorff dimension of D is 3n-1/2.


Victor Donnay (Bryn Mawr)

Embedded surfaces with Anosov geodesic flow.

Abstract. We construct surfaces, isometrically embeddable in R3, whose geodesic flow is Anosov. It is known that no metric on surfaces of genus 0 or 1 can have an Anosov geodesic flow, while for surfaces of genus 2 or higher there exist metrics of negative curvature that produce Anosov flows, but these negatively curved metrics can not be isometrically embedded.

Our construction produces surfaces with Anosov geodesic flow for all sufficiently large genus, but does not produce any explicit bounds on the genus. An open question is whether for all genus above 1, there exist embeddable metrics with Anosov geodesic flow or, if not, to give lower bounds on the genus. This result is joint work by V.J. Donnay and C. Pugh and builds on their earlier work of constructing embedded surfaces with Bernoulli geodesic flow using the finite horizon cap construction.


Tomasz Downarowicz (University of Maryland)

Entropy properties in topological dynamics.

Abstract. In a non-uniquely ergodic system (action of a single homeomorphism T) the seemingly most complete information about entropy is in the entropy function (assigning to each invariant measure the entropy of T with respect to this measure). However, for many aspects like asymptotic h-expansiveness, existence of a subshift cover, etc., we need a more subtle tool, namely we need to know how the entropy function is approximated by certain relative entropy functions. We will present some recent results concerning the entropy function, relative entropy functions, relative variational principles, existence of subshift covers, criteria for asymptotic h-expansiveness, and the like. We will propose a new topological invariant (which we call "entropy structure") carrying complete information about the above discussed entropy properties of the system.


Renato Feres (Washington University)

Foliated Liouville theorems.

Abstract. It is an elementary and well-known fact that a holomorphic function on a compact complex manifold without boundary is constant. In the setting of compact foliated spaces with (not necessarily compact) complex leaves, it becomes necessary to take into account the foliation dynamics in order to understand whether and to what extend continuous leafwise holomorphic functions are leafwise constant. The talk will describe a number of results in this direction due to A. Zeghib and the speaker.


David Fisher (Yale University)

Local rigidity of partially hyperbolic and isometric lattice actions.

Abstract. Let G be semisimple Lie group with all simple factors noncompact and of real rank at least 2. Let $\Gamma$ be a lattice in G. I will discuss the proof of the following:

THEOREM: Any standard affine action of G or $\Gamma$ on a compact manifold is locally rigid.

By a standard affine action we mean any affine action on a compact homogeneous space of the form $H/{\Lambda}$ where $\Lambda$ is discrete and cocompact. By local rigidity of an action $\rho$ we mean that any other action $\rho'$ that is close to $\rho$ on a compact generating set (in an appropriate topology on diffeomorphisms) is conjugate to the original action.

We also prove the theorem for some more general actions. Let M be a compact manifold with an isometric $\Gamma$ action and let $\Gamma$ act affinely on $H/{\Lambda}$ as above. Then the diagonal action of $\Gamma$ on $M{\times}H/{\Lambda}$ is also locally rigid.

The results are primarily new in the case where the action is partially hyperbolic. The only previous results in the partially hyperbolic setting require strong assumptions on the central leaves of the action where we need none.

This is joint work with G.A. Margulis.

 

Arek Goetz (San Francisco State University)

Examples of rational piecewise rotations with unbounded return times.

Abstract. In a joint work with Gauillaume Poppaggalla we show that there are examples of rational piecewise rotations with three atoms whose return times to one of the atoms, though, by Poincare result, almost everywhere finite, are not uniformly bounded. The examples are based on the angle of rotation $\pi/7$. The talk will be accessible to non experts in the area and it will be augmented by a short computer presentation.


Boris Gurevich (Moscow State University and Penn State)

Multifractal analysis of ergodic averages for multidimensional time parameter.

Abstract. This is a joint paper with Arkady Tempelman. Let X be the space of all functions on $\mathbb Z^d,\,d\ge1$, with values in a finite set. Let $\tau_s,\,s\in\mathbb Z^d,$ be the translation group on X and $T_n=[-n,n]^d\cap\mathbb Z^d$. For continuous real-valued functions $f_1,\dots,f_m$$(m\ge1)$ and for any $a_1,\dots,a_m\in\mathbb R$ we evaluate the Hausdorff dimension of the set

 

\begin{displaymath}\left\{x\in X:\lim_{n\to\infty}\frac{1}{(2n)^d}\sum_{s\in T_n} f_i(\tau_sx)=\alpha_i,\,i=1,\dots,m\right\}\end{displaymath}

 

in terms of a Statistical Physics model related to $f_1,\dots,f_m$. We also evaluate the Hausdorff dimension of the set of generic points for translation invariant Gibbs measures. The main challenge here is due to the possibility of phase transitions when $d\ge1$. Our proof is based on a generalization of a method invented by Cajar.


Nicolai Haydn (University of Southern California)

A Central Limit Theorem for maps that are $(\phi,f)$ mixing.

Abstract. We prove that the Central Limit Theorem applies to the log of the measure of cylindersets for measures that are $(\phi,f)$-mixing. We approximate the measure of small cylinders by products of smaller cylinders and use the mixing property to show that in this way we get `nearly independent' random variables. Moreover, we show that the variance obtained from the variance of the information function and the entropy of joins. As a corollary we prove that the repeat time satisfies a CLT. This result involves the exponential law of the limiting distribution of returns to cylinder sets. Previously such results had been shown for Gibbs measures where the decay of correlations was used to obtain independence in the limit.


Huyi Hu (University of Southern California)

Absolutely continuous invariant measures for non-uniformly expanding maps.

Abstract. This is a joint work with Sandro Vaienti. We consider some multidimensional piecewise smooth expanding maps with an indifferent fixed point that may not have Markov partition. We show that such systems admit absolutely continuous invariant measures $\mu$ that has at most finitely many ergodic components, which are either finite or sigma-finite. Also, examples show that $\mu$ may has both finite and sigma-finite ergodic components simultaneously, and both contain the indifferent fixed point in their supports.


Jarek Kwapisz (Montana State University)

Rotation sets and combinatorics of torus diffeomorphisms.

Abstract. The rotation sets of isotopic to the identity diffeomorphisms of the two torus are convex compact subsets of the plane. What sets are realized is not fully understood. Also, even if the rotation set is just a single non-resonant irrational vector, little is known about the map (besides the local KAM type results).

I will present two theorems. The first asserts that the rationally sloped segments that contain no rational points and some irrationally sloped segments cannot be rotation sets (unless they degenerate to points). The second secures existence of renormalization and dynamical partitions of the torus and shows that all maps with a fixed non-resonant irrational vector for their rotation set are combinatorially equivalent.


John Mather (Princeton University)

Arnold diffusion.

Abstract. I will describe work in progress concerning the existence of Arnold diffusion in small Hamiltonian perturbations of integrable systems with positive definite normal torsion. Considering the unperturbed integrable system as the origin in a function space, we show the existence of three open sets U, V, and W in the function space of Hamiltonian systems. The set U is non-void and positively homogeneous. The set V contains an initial segment of every ray r in U emanating from the origin. The set W is open and dense in V. A system in W exhibits Arnold diffusion.


Zbigniew Nitecki (Tufts University)

Preimage entropy and symbolic dynamics.

Abstract. This is a joint work with Doris Fiebig and Ulf-Rainer Fiebig. Topological entropy htop is based on the dispersion of orbits in forward time. For noninvertible maps, several conjugacy invariants based on preimage structure have been formulated and studied. Pointwise preimage entropy hm is the growth rate of the maximum number of $(n,\epsilon)$-separated nth preimages of a point in the space, while branch preimage entropy hb is the growth rate of the number of $(n,\epsilon)$-distinct preimage sets of points.

Hurley showed that $h_m\leq h_{top}\leq h_m + h_b$, and examples show that either inequality can be strict. For a (one-sided) subshift, hm is the growth rate of the size of the largest "nth predecessor set" while hb is the growth rate of the number of distinct such sets.

Theorem 1: For a subshift (and more generally for a forward-expansive map on a compact metric space), hm=htop and there exists an entropy point: a point for which the number of nth preimages grows at precisely this rate. A more general notion of "entropy point" can be formulated, and such points can be shown to exist for any asymptotically h-expansive map. However, we construct an "almost" h-expansive map with no entropy point--in fact, for this map the number of $(n,\epsilon)$-separated nth preimages of any particular point grows subexponentially, but hm>0. (This answers a question raised by Hurley.)

A second result concerns inverse limits (or natural extensions). It is well known that non-conjugate systems (even shifts) can have conjugate inverse limits.

Theorem 2: Forward-expansive surjections on compact metric spaces (in particular subshifts) whose inverse limits are conjugate have equal branch preimage entropy.

 

Viorel Nitica (West Chester University)

Transitivity of euclidean extensions of Anosov diffeomorphisms.

Abstract. Let X be an infranilmanifold, $T:X\to X$ an Anosov diffeomorphism, $f:X\to{\mathbb R}^n$ a Holder function, and Tf the skew-product determined by T and f. Then the following are equivalent:

1)
Tf is topologically transitive;
2)
Tf is stably topologically transitive;
3)
Tf has orbits with projections arbitrarily large in any direction;
4)
Tf satisfies the following Inseparability Hypothesis: the weights of Tf over the periodic points are separated by any hyperplane.


Hee Oh (Princeton University)

Hecke operators and equi-distribution of integer points on a family of homogeneous varieties.

Abstract. Let f be a homogeneous polynomial with integer coefficients, and let Vm be the variety defined by f=m. In the early sixties, Linnik raised the problem of understanding the distribution of the integer points Vm(Z) as m tends to infinity. In complete generality it seems hopeless to attack this question, except when the number of variables of f is much bigger than the degree of f in which case the Hardy-Littlewood circle method can be applied.

In this talk we discuss Linnik's problem when f arises from invariant theory, explaining how the Hecke operators then play a role here. (joint work with W. T. Gan).


Rodrigo Perez (SUNY Stony Brook)

Finite recurrence patterns in the quadratic family.

Abstract. The principal nest is a collection of pieces in the Yoccoz puzzle of a Julia set. It encodes properties of the critical orbit that may imply strong geometric results like local connectivity and hairiness. In the talk we endow the principal nest with extra structure that completely determines the combinatorial class of the map. One important consequence is a complete characterization of complex quadratic Fibonacci maps. This is done as a particular case of a large family of examples with simple recurrence.


Federico Rodriguez-Hertz (IMPA)

Stable ergodicity of some linear automorphisms of the torus.

Abstract. It is proved that some linear automorphisms of ${\mathbb T}^N$ are stably ergodic. The ones we deal with include all ergodic linear automorphisms when N=4.


Nándor Simányi (University of Alabama)

The ergodicity of typical hard sphere systems in 2D: geometric aspects.

Abstract. In my recent proof of the Boltzmann-Sinai Ergodic Hypothesis for typical hard sphere systems (in 2D) I used some interesting geometric arguments, such as

1) the possibility of an infinite, neutral, singularity-free translation (other than in the direction of the flow!) of any phase point $x_\infty$that is actually an $\omega$-limit point of the trajectory of a separating phase point (in any neighborhood of which one finds two ergodic components);

2) the classical Sylvester-Gallai theorem from combinatorial geometry, claiming that for any finite subset X of the Euclidean plane, if no line intersects X in exactly two points, then X is contained by a single line.

The emphasis of my talk will be put on these geometric aspects.


Michael Shub (IBM) Mathematics Department Colloquium

Order out of Chaos? Some recent examples.

Abstract. One of the hallmarks of chaotic dynamical systems is that the long term future is difficult to predict deterministically because it depends very sensitively on initial conditions. In these circumstances one may still be able to make statistical predictions. Some of the earliest results in this direction are Hopf's theorem on the ergodicity of the geodesic flow on compact surfaces of constant negative curvature in the 1930's and Anosov's generalization on the ergodicity of uniformly hyperbolic systems in the 1960's. Here we discuss some recent generalizations of the theorems of Hopf and Anosov and the resolution of some long standing problems by Dolgopyat-Pesin and Rodriguez-Hertz. Some comments will be made contrasting the topological classification and ergodic theory of these systems. I intend the talk to stay on an elementary level with the main examples given by linear automorphisms of tori. Let Abe an element of $SL(n,\mathbb Z)$ the n by n matrices with integer entries and determinant one. Then A defines a linear map of ${\mathbb R}^n$. Since the integer lattice ${\mathbb Z}^n$ is preserved by A, A induces a diffeomorphism of the n-torus, ${\mathbb T}^n$, where ${\mathbb T}^n$ is considered as ${\mathbb R}^n$ mod ${\mathbb Z}^n$.

 

Domokos Szász (Technical University of Budapest)

Algebraic methods and the ergodic hypothesis for hard balls.

Abstract. The Ergodic Hypothesis for Hard Balls says that the systems of N elastic hard balls moving on the d-torus is ergodic modulo the trivial invariants of motion. A simple observation is that this system is isomorphic to a billiard with convex obstacles (which are even strictly convex if N=2). For N=2 the hypothesis got settled by the Moscow school (Sinai, 1970, d=2; Chernov-Sinai, 1987, d > 2). Afterwords the Budapest school reached various results for N >2 (most notably Krámli, Simányi and Szász for N=3 and 4 and Simányi for d > N-1) by introducing dynamical-topological and geometric-algebraic methods.

In 1999, Simányi and Szász gave a partial solution of the Boltzmann-Sinai Ergodic Hypothesis by establishing that typical hard ball systems are hyperbolic, i. e. all their Lyapunov exponents are non-zero a. e.. Several experts, however, were surprised that their methods relied quite heavily on the fact that in the isomorphic billiard system the boundaries of the obstacles are quadratic algebraic manifolds (indeed, they are cylinders with spherical bases). Earlier, in the theory of hyperbolic billiards only some smoothness, C2 in general, of these boundaries was assumed and used.

Nevertheless, more recently it turned out that the algebraicity of the obstacle boundaries seems to be an essential assumption even in the fundamentals of the theory of billiards with convex obstacles. In fact, here a useful property of algebraic submanifolds is that they can be decomposed into a finite number of nice Lipschitz graphs of functions (these recent results are joint with P. Bálint, N. Chernov and P. Tóth).

 

Sergei Tabachnikov (Penn State)

Multi-dimensional Birkhoff theorem: periodic trajectories in smooth convex billiards.

Abstract. The classical Birkhoff theorem concerns periodic trajectories in smooth strictly convex plane billiards; it asserts that for every n and $r \leq n/2$ there exist two distinct n-periodic billiard trajectories with rotation number r. In a recent series of papers, M. Farber and myself found generalizations of this result in multi-dimensional setting. One of the main results reads: for a generic smooth strictly convex billiard table in m-dimensional space, there exist at least (n1)(m1) distinct n-periodic billiard trajectories. The work is based on a topological study of the cyclic configuration space of the sphere and Morse-Lusternik-Schnirelman theory.


Thomas Ward (University of East Anglia)

A mixing rigidity result for ${\mathbb Z}^d$ actions.

Abstract. A conjectured rigidity result for higher-order mixing properties of algebraic ${\mathbb Z}^d$-actions is described, with proof in special cases. This is joint work with Dr. Manfred Einsiedler.


Alistair Windsor (Penn State)

Mixed spectrum reparameterizations of linear flow on ${\mathbb T}^2$.

Abstract. We consider time changes of an irrational flow on ${\mathbb T}^2$ defined by $ \frac{dx}{dt} = \alpha \qquad \frac{dy}{dt} = 1$. The study of these time changes began with Kolmogorov and many of the basic questions in the area date from his 1954 I.C.M. address. The reparameterized flow remains minimal and uniquely ergodic but may exhibit other ergodic properties which are quite distinct from the original linear flow. That a plethora of ergodic properties can be obtained via continuous time changes follows from Kakutani equivalence. For sufficiently smooth reparameterizations the situation is more subtle and depends on the arithmetic properties of $\alpha$. For Liouville $\alpha$ the a generic $C^\infty$ time change results in a flow which is weak mixing. This is in stark contrast to the original linear flow which has pure point spectrum.

In joint work with B. Fayad [C.N.R.S.] and A. B. Katok we prove that for any Liouville flow there exist $C^\infty$ time changes for which the resulting flow has mixed spectrum.


Todd Young (Ohio State University)

Absolutely continuous invariant measures and saddle-node bifurcations.

Abstract. We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. It was previously shown that for a parameter set of positive Lebesgue density at the bifurcation, the maps possess attracting periodic orbits of high period. We show that there is also a parameter set of positive density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to an atomic measure supported on the orbit of the saddle-node point. Using these measures we analyze the intermittent time series that result from the destruction of the periodic attractor in the saddle-node bifurcation and prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the periodic attractor.


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