Abstracts | PSU Mathematics Department

DYNAMICAL SYSTEMS and RELATED TOPICS WORKSHOP

October 26-29, 2000
Penn State
Atherton Hotel
State College, PA

ABSTRACTS OF TALKS

Jose Alves (University of Maryland and University of Porto, Portugal), Nonuniformly expanding dynamics: Stability from a probabilistic viewpoint

Abstract. We present some recent developments on the theory of smooth dynamical systems exhibiting nonuniformly expanding behavior. In particular, we show that these systems have a finite number of SRB measures whose basins cover the whole ambient space, and under some conditions on the rate of expansion their dynamics is statistically stable.

Eric Bedford (Indiana University at Bloomington), Complex dynamics and real horseshoes I

John Smillie (Cornell University), Complex dynamics and real horseshoes II

Abstract. We will discuss some recent results on the dynamics of polynomial diffeomorphisms of C². As a consequence of this work, we will derive some results about the boundary of the horseshoe locus in the Henon family of diffeomorphisms of R², and we will discuss some open questions.

Keith Burns (Northwestern University), Magnetic flows (joint work with Gabriel Paternain)

Abstract. I will talk about how the topological entropy of the magnetic flow changes as the intensity of the magnetic field increases.

There is a lower bound for the topological entropy which is comparable to Manning's lower bound for the entropy of the geodesic flow: it is the growth rate of the average volume of certain balls defined using minimizing magnetic geodesics. This lower bound is nonincreasing as the intensity increases, and is equal to the topological entropy when the magnetic flow is Anosov.

These results are complemented by examples in which the magnetic flow is Anosov for low values of the intensity and becomes non-Anosov and then Anosov again as the intensity increases. Moreover it is possible for the topological entropy to increase when flow becomes non-Anosov.

Chris Connell (University of Illinois at Chicago), Lyapunov exponent rigidity for nonpositively curved manifolds

Abstract. We answer a question posed by Ralf Spatzier at the 1999 AMS Summer Institute on smooth ergodic theory. We show that for an irreducible nonpositively curved manifold if almost every vector in its unit tangent bundle possesses a minimal (determined by the curvature) Lyapunov exponent for the geodesic flow then it is locally symmetric of noncompact type. We also discuss implications.

Gregory Galperin (Eastern Illinois University), Illumination of a convex polygon by a billiard search light

Abstract. A long standing problem (from 1958) on illumination of a non-convex polygonal room (polygon with mirror sides) by a source of lights emanated in all possible directions (360 degrees) had been solved negatively several years ago. A Canadian mathematician Tokarsky constructed an example of a non-convex polygon that is not illuminable from a particular point on the table: the light rays can not reach some other table's point. What would be the answer to the same problem if the table is a convex polygon but the source of lights is now a search light which illuminates an arbitrary small angle $\epsilon>0$ ? The speaker posed this question at the AMS conference in 1999. The talk will be devoted to the speaker's solution to the problem for a square and to the discussion of other convex polygons.

Alex Haro (Univ. of Texas, Austin), Invariant manifolds in quasi-periodic maps: rigorous results and computations (joint work with Rafael de la Llave)

Abstract. We explain the parameterization method to prove the existence of invariant manifolds in quasi-periodic systems. The method not only gives a proof of the usual stable and unstable manifolds, but also of non-resonant ones. It also provides an effective algorithm to compute these manifolds. We display the numerical results for the quasi-periodic forced Henon map and standard map.

Nicolai Haydn (University of Southern California), Distribution of return times for dynamical systems

Abstract. We develop a general framework that allows to prove that the limiting distribution of return times are Poisson distributed. The approach uses a result that connects the convergence of factorial moments to the mixing properties of transformations which often times are expressed through the decay of correlations. With this method we can prove distribution results in a variety of settings. A number of our results generalise previous distribution results to a wider class of maps or provide much improved error estimates. For $\phi$ -mixing maps we obtain a close to exhausting description of return times. For $(\phi,f)$ -mixing maps we can show how the separation function affects error estimates for the limiting distribution. Examples of $(\phi,f)$ -mixing maps are piecewise invertible maps and rational maps. These distribution results can be used to obtain estimates for large deviations.

Suzanne Lynch Hruska (Cornell University), Hyperbolicity in the complex Henon family

Abstract. The Henon map,

$H_{a,c}: (x,y) \rightarrow (x^2+c-ay, x)$

is a widely studied family of maps with complicated dynamical behavior. H has been studied as a diffeomorphism of R² with a,c real parameters. Here we regard H as a holomorphic diffeomorphism of C², and allow a, c to be complex. The dynamics of H restricted to R² are contained in the dynamics of H on C², so we can learn about the first map by using tools of complex analysis. Recall, a map is hyperbolic over a set X if for each x in X there is a splitting of the tangent bundle over x into an ``unstable'' and a ``stable'' direction, with the unstable direction uniformly expanded by the map and the stable direction uniformly contracted. A complex Henon map is called hyperbolic if it is hyperbolic over its Julia Set, J, which is the boundary of the set of all points whose orbits under H and H^-1 are bounded. We have developed and implemented a computer algorithm to test whether for a given a,c, the map H_a,c is hyperbolic. If the program outputs ``yes'', then the computation proves the map is hyperbolic. If the answer is ``no'', the test was inconclusive. This talk will sketch the algorithm, and give some initial applications.

Brian Hunt (University of Maryland), A robust example (?) of strange nonchaotic dynamics (Joint work with Edward Ott)

Abstract. In the last few years, scientists have become increasingly interested in "strange nonchaotic" attractors in the class of quasiperiodically forced dynamical systems (skew-product systems with quasiperiodic base). Do such attractors appear robustly in this class of systems or do they occur only for atypical systems? By "nonchaotic" we mean that the attractor has no positive Lyapunov exponents, and by "strange" we mean that the attractor is geometrically complex - to be more precise, that it supports a natural
invariant measure whose dimension is less than the topological dimension of the attractor. For the most part this phenomenon has been observed numerically, and examples in which strange nonchaotic behavior has been established rigorously require a symmetry in the fiber of the skew-product system. We describe a general class of skew-product diffeomorphisms of the 2-torus, whose base is an irrational rotation of the circle, and argue based on a combination of rigorous results and numerical evidence that
they do robustly exhibit strange nonchaotic dynamics. Specifically, we prove that these systems are topologically transitive yet we conjecture that their only invariant measures have Hausdorff dimension 1.

Steve Hurder (University of Illinois at Chicago), Dynamics of group actions on the circle

Abstract: We will discuss three complementary results about the dynamics and properties of a group acting on a circle. The first is a simple proof of a result answering a question posed by Tom Ward in summer 1999:

Theorem: If a discrete group G acts by homeomorphisms on the circle, and the action is expansive, then G has a free non-abelian sub-semigroup with a resilient (homoclinic) orbit, and hence the action has positive entropy.

The second solves a problem posed by Ghys in summer 1998, and recently proved by Margulis:

Theorem: If a discrete group G acts by homeomorphisms on the circle, and there is no invariant measure, then G has a free non-abelian subgroup.

The third is a new proof of an old result of Ghys, Langevin and Walczak for C²actions. The new proof is purely dynamical, using a version of "Pesin Theory" for C¹ actions of groups and foliations:

Theorem: If a discrete group G acts by C¹diffeomorphisms on the circle, and the action has positive entropy, then G has a free non-abelian sub-semigroup with a resilient (homoclinic) orbit.

The proofs of all three theorems share many similar techniques, which give new understanding of the dynamics of group actions on the circle.

Julij Ilyashenko (Cornell University), Upper estimates of the number of limit cycles for Abel and Lienard equations (partly based on a joint work with A. Panov)

Abstract. We estimate the number of limit cycles of planar vector field through the size of the domain of the Poincaré map, the increment of this map and the width of the complex domain to which the Poincaré map may be analytically extended. The estimate is based on the relation between growth and zeros of holomorphic functions. This estimate is then applied to getting the upper bound of the number of limit cycles of Lienard equation

through the (odd) power of the monic polynomial F and magnitudes of its coefficients. In the same way, an upper bound of the number of limit cycles of the Abel equation is obtained.

Francois Ledrappier (Northwestern University and CNRS), Linear actions of 2 x 2 complex matrices

Abstract.We consider the linear action of a group $\bar \Gamma$ of 2 x 2 matrices on C², where $\bar \Gamma$ is a normal subgroup of an uniform lattice $\Gamma$ in SL(2,C), with quotient isomorphic to Z^d. We obtain results on invariant measures and asymptotic distribution of orbits. The proof is based upon symbolic dynamics for the frame flow on hyperbolic 3-dimensional manifolds. This is a joint work with Mark Pollicott.

Alexander Leibman (Ohio State University), Polynomial unitary actions of nilpotent groups

Abstract. A mapping F from a group G into a group H is called polynomial if it trivializes after several "differentiations" of the form DF(g)=F(hg)/F(g). A polynomial unitary action of a group G on a Hilbert space M is a polynomial mapping from G into the group of unitary operators on M. We show that in many respects the polynomial unitary actions of nilpotent groups behave like the conventional unitary actions. This leads to various ergodic-theoretical applications.

Elon Lindenstrauss (IAS), Bernoulli convolutions and a question of Sinai on attainable entropies (joint work with Yuval Perez and Wilhelm Schlag)

Abstract. We use Bernoulli convolutions to study certain linear realizations of a Bernoulli process on two symbols as a real valued stationery process, which we show has trivial left and right tails. New properties of Bernoulli convolutions and more general projected measures in the symbolic context are established. These properties are substantially stronger and more general than is required for the study of the linear realizations. This construction is used to answer a question of Sinai regarding the possible values of the entropy of K-partitions for general Bernoulli measure preserving systems.

Howard Masur (University of Illinois at Chicago), Asymptotic formulas for the number of saddle connections on flat surfaces

Abstract.This represents joint work with Alex Eskin and some of it is joint with Anton Zorich. Suppose Sis a flat surface with isolated cone singularities. Such surfaces arise in the study of rational billiards, quadratic and abelian differentials on compact Riemann surfaces and are natural generalizations to higher genus of a flat torus. A saddle connection is a geodesic joining two singularities. We are interested in the asymptotics of the number of saddle connections of length less than Tas T goes to infinity. We show that for a "generic" surface the asymptotics are cT² for a constant c that depends only the topology of the surface and not the flat structure itself. We then show how to compute the constant c. We also consider the related problem of counting the number of closed geodesics that do not pass through a singularity.

John Mather (Princeton University), Differentiability of the minimal average action in one higher dimension

Abstract. We suppose that L satisfies the Legendre condition, has fiberwise superlinear growth, and that the Euler-Legendre flow is complete. Let $\beta:H_1(M,{\mathbb R})\to{\mathbb R}$ be the minimal average action. In the case $\dim M=1$ , Bangert showed that $\beta$ is differentiable at irrationals. We generalize this result to $\dim M\leq 2$ . Burago, Ivanov, and Kleiner have shown that such a result cannot be generalized to arbitrary dimensions, at least in the case of finite differentiability of the Lagrangian. The case of infinite differentiability remains open.

Roman Muchnik (Yale University), Semigroup actions of GL(n,Z) on Tⁿ

Abstract. For large semigroups for GL(n,Z) the orbit of every non-rational point is dense in n-dimesional torus T^n. I will discuss the properties of such semigroups. This generalizes the work of Furstenberg for a circle for multipication by 2 and 3 and the work of Berend for commutatives semigroups on n-dimensional torus.

Viorel Nitica (University of Notre Dame), Stable topological transitivity of skew-products over hyperbolic sets

Abstract. We show that the set of Hölder continuous skew-products by a compact connected semisimple Lie group, over a hyperbolic basic set, contains an open dense set of topologically transitive transformations. As a corollary of our result it follows that the set of Hölder skew-products by a compact connected semisimple Lie group, over a basic hyperbolic set, contains an open dense set of ergodic transformations.

Leonid Polterovich (Tel-Aviv University), Kick stability in groups and dynamical systems

Abstract. "How far can a flow be kicked?" More precisely, consider the behavior of a dynamical system under the influence of a sequence of kicks arriving periodically in time. We are interested in the following stability type question: does the kicked system inherit some recurrence properties of the original flow? It turns out that in some situations (linear flows on tori, cat maps, fastly growing Hamiltonian flows on symplectic manifolds) such a stability indeed takes place even when the kicks are quite large. The talk is based on a joint work with Zeev Rudnick.

Enrique Pujals (Catolic University of Rio de Janeiro and Courant Institute), Partially hyperbolic systems and robust transitivity

Abstract. We will give some results about partial hyperbolicity and robusttransitivity.

Martin Sambarino (University of Maryland), On surface diffeomorphisms

Abstract. We will give some results on the global dynamics of surface diffeomorphisms based on the study of the Dominated Splitting.

Joerg Schmeling (Free University, Berlin), Diophantine numbers, dimension and Denjoy maps

Abstract. We study the effect of the arithmetic properties of the rotation number on the minimal set of an aperiodic, orientation preserving diffeomorphism of the circle. The box dimension and the Hausdorff dimension may differ. The box dimension depends only on the smoothness of the map while the Hausdorff dimension depends on both the Diophantine class of the rotation number and on the smoothness of the map.

Enrico Valdinoci (Univ. of Texas, Austin), Speed of Arnold diffusion

Abstract. In relation to the problem of Arnold diffusion, we consider several nearly integrable Hamiltonian systems and we show that the diffusion time is of the order of a power of the homoclinic splitting. The proof is based on a combination of KAM results and Mather theory.

Peter Veerman (CUNY-Queens College and Portland State University), Furstenberg's conjecture

Abstract. Furstenberg has conjectured that the invariant set of certain affine iterated function system, whose geometry cannot be described by a finite directed graph must have Hausdorff dimension=similarity dimension. We discuss some partial results and some numerical experiments.

Amie Wilkinson (Northwestern University), C¹ density of stable accessibility

Abstract. This is work with Dima Dolgopyat.

A volume-preserving, ergodic C²-diffeomorphism is stably ergodic if it remains ergodic after a sufficiently C¹-small volume preserving perturbation. A conjecture of Pugh and Shub relates stable ergodicity to partial hyperbolicity. A diffeomorphism $f:M\to M$ is partially hyperbolic if the tangent bundle to M breaks into three invariant subbundles, one expanding, one contracting, and the third neither expanded nor contracted as strongly as the other two.

Conjecture 1: For $r\geq 1$ , the stably ergodic C^r-diffeomorphisms are dense among the partially hyperbolic diffeomorphisms.

A closely related conjecture concerns the basic property of stable accessibility. A partially hyperbolic diffeomorphism $f\in \hbox{Diff}(M)$ is stably accessible if any 2 points in M can be connected by a path staying tangent to the leaves of the stable and unstable foliations associated to f.

Conjecture 2: For $r\geq 1$ , the stably accessible C^r-diffeomorphisms are dense among the partially hyperbolic diffeomorphisms.

Pugh and Shub proved that Conjecture 2 implies Conjecture 1 under (relatively mild) additional hypotheses. We give a proof of Conjecture 2 in the C¹ case; in fact we prove C¹-density for C^rdiffeomorphisms for all $r\geq 1$ .

James Yorke (IPST, University of Maryland), Smooth embedding of non-smooth sets

Abstract. Takens, Ruelle, and Eckmann launched an investigation of images of attractors of dynamical systems. If a compact set A is an attractor in Rⁿ and g: Rⁿ -> R^m is a generic smooth map, and if m < n, how do A and g(A) compare? By making physical measurements, physics in effect are examining a set g(A) and would like to recover as many properties of A by examining g(A), hence the need for understanding how they compare. If A is a manifold, the problem is relatively easy, but that is rarely the case if the dynamics are chaotic. When A is chaotic, the properties studied include the dimension of A or the Lyapunov exponents.

Robert Yuncken (Penn State), Regular tessellations of hyperbolic space by fundamentalregions of a Fuchsian group

Abstract. It is well-known that for every p,q with 1/p + 1/q < 1/2, the hyperbolic plane admits a tessellation by regular p-gons with q polygons meeting at each vertex which is called regular tessellation of type {p,q}. In this short talk I will present a necessary and sufficient condition for such a tessellation to be realizable as a tessellation by fundamental domains of some Fuchsian group. Namely, the tessellation of type {p,q} is a tessellation by fundamental domains if and only if q has a prime divisor less than or equal to p.