PSU Mark
Eberly College of Science Mathematics Department

Abstracts

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DYNAMICAL SYSTEMS and RELATED TOPICS WORKSHOP

October 12-15, 2002
Penn State

ABSTRACTS OF TALKS



Alain Chenciner (Paris VII)

Continuing the ``Eight'' in 3-space: Marchal's P12 family

Abstract. Christian Marchal proposed a modification of the variational characterization of the ``Eight'' solution of the three-body problem, which should lead to a family of spatial choreographies with the same 12-fold symmetry. The mathematical status of this family will be discussed.


Wen-Chiao Cheng (Michigan State)

Relations among conditional entropy, topological entropy and pointwise preimage entropy

Abstract. In this talk we first introduce new properties of pointwise preimage entropy which are defined by Hurley, Nitecki and Przytycki. Later we define new conditional topological and measure-theoretic preimage entropies and study their properties. Among other things we obtain analogs of the variational principle and Shannon-McMillan-Breiman theorem for these new invariants.


Dmitry Dolgopyat (Institute for Advanced Study)

Stable ergodicity of random rotations

Abstract. Let f and g be area preserving smooth diffeomorphisms of the two dimensional sphere which are close to rotations P and Q respectively. If P and Q topologically generate SO(3)then any set invariant with respect to both f and g has measure 0 or 1. This is a joint work with Rafael Krikorian.


Patrick Eberlein (University of North Carolina)

Structure of 2-step nilpotent Lie algebras of type (p,q) 

Abstract. For pairs of integers $p \ge 1$, $q \ge 2$ let N(p,q)denote the set of 2-step nilpotent Lie algebra structures on $\mathbb{R} ^{p+q}$ that have a derived algebra $[\mathbb{R} ^{p+q}, \mathbb{R} ^{p+q}]$ of dimension p and codimension q. The space N(p,q) is a smooth connected manifold of dimension pq + pD, where D =(1/2)q(q-1). Moreover, N(p,q) is also a fiber bundle over the Grassmann manifold G(p,p+q) of p-dimensional subspaces of $\mathbb{R} ^{p+q}$. There is a natural action of $GL(p+q,\mathbb{R} )$ on N(p,q) whose orbits are the isomorphism classes of N(p,q).

Let $so(q,\mathbb{R} )$ denote the real $q \times q$ skew symmetric matrices, and let $GL(q,\mathbb{R} )$ act on $so(q,\mathbb{R} )$ by g(A) = gAg*, where g* denotes the transpose of g. Let G(p,so(q,R)) denote the Grassmann manifold of p-dimensional subspaces of $so(q,\mathbb{R} )$. The action of $GL(q,\mathbb{R} )$ on $so(q,\mathbb{R} )$ extends naturally to an action of $GL(q,\mathbb{R} )$ on $G(p,so(q,\mathbb{R} ))$.

Theorem. The quotient space $X(p,q) = N(p,q) / GL(p+q,\mathbb{R} )$ of isomorphism classes in N(p,q) is homeomorphic to the compact quotient space $G(p,so(q,\mathbb{R} )) / GL(q,\mathbb{R} )$.

Corollary. (Duality) X(p,q) is homeomorphic to X(D-p,q) for all integers $p \ge 1$, $q \ge 2$, where D = (1/2)q(q-1).

Generic local rigidity. We define the dimension of X(p,q) to be the smallest codimension of an orbit of $GL(p+q,\mathbb{R} )$ in N(p,q), or equivalently, the smallest codimension of an orbit of $GL(q,\mathbb{R} )$ in $G(p,so(q,\mathbb{R} ))$. The cases where X(p,q) has dimension zero, or equivalently where $GL(p+q,\mathbb{R} )$ has open orbits in N(p,q), is of special interest. Every element $\{R^{p+q}, [\; ,\;]\}$ in an open orbit of $GL(p+q,\mathbb{R} )$ is locally rigid; that is, $\{\mathbb{R} ^{p+q}, [\; ,\; ]\}$is isomorphic to all elements $\{\mathbb{R} ^{p+q}, [\; , \;]'\}$ in some open set of N(p,q)that contains $\{R^{p+q}, [\; ,\;]\}$. The union of all open GL(p+q,R) orbits in N(p,q) is a dense open (possibly disconnected) subset of N(p,q).

Spaces of dimension zero. Up to the duality $(p,q)\rightarrow (D-p,q)$ the following is a complete list of pairs for which X(p,q) has dimension zero: [1] (1, 2k) (Heisenberg algebras); [2] (D, q), $q \ge 2$ (free 2-step nilpotent Lie algebras); [3] (2, 2k+1); [4] (2, 4); [5] (2, 6); [6] (3,4); [7] (3, 5); [8] (4, 5). Examples [1] through [5] were previously well known.

Dimension of X(p,q). The dimension of X(2,2k) is k-3 for $k \ge 4$. Except for this case and the cases of dimension zero listed above X(p,q)has dimension 1 + p(D-p) - q2.

Lattices. In another direction we describe some highly nongeneric examples in X(p,q) that arise from p-dimensional subalgebras and Lie triple systems in $so(q,\mathbb{R} )$. All of these examples of 2-step nilpotent Lie algebras admit a rational structure, and hence the corresponding simply connected, 2-step nilpotent Lie groups admit a lattice by the criterion of Mal'cev.

 

Albert Fathi (ENS Lyon)

The interface of PDEs and Lagrangian dynamics: existence of C1 subsolutions

Abstract. This is a joint work with Antonio Siconolfi (La Sapienza, Roma).

We will discuss for Lagrangian Dynamical Systems (satisfying the usual Mather's condition), the relevance of subsolutions of the corresponding Hamilton-Jacobi equation.

The main result is to show that there is a C1 subsolution of u satisfying $H(x,d_xu)\le c[0]$, where c[0] is the Mañé critical value. This settles a ``calibration'' problem related to the Aubry(-Mather) set, that has been studied, among others, by Ricardo Mañé in his last papers. It has also consequences on the dynamics of the Lagrangian System.

The proof is via some deep connections of the Aubry(-Mather) set, and viscosity solutions of Hamilton-Jacobi equations. This connection will also explain why it is impossible to obtain C2 subsolutions.


Basam Fayad (CNRS)

Elliptic constructions and nonuniform hyperbolicity

Abstract. We combine elliptic constructions and perturbations of partially hyperbolic systems to obtain some non standard examples of nonuniformly hyperbolic systems.


David Fisher (Lehman College)

Continuous and smooth orbit equivalence rigidity

Abstract. Let $\Gamma_1$ and $\Gamma_2$ be finitely generated groups and $\rho_1$ and $\rho_2$ actions of $\Gamma_1$ and $\Gamma_2$ on compact manifolds spaces M1 and M2. We call a Ck diffeomorphism $f:M_1{\rightarrow}M_2$ a Ck orbit equivalence if it maps orbits to orbits. We call $\rho_1$ Ck orbit equivalence rigid if any Ck orbit equivalence is equivariant.

We prove that the natural $SL_n(\mathbb Z)$ action on ${\mathbb T}^n$is C1 orbit equivalence rigid and that the natrual ${\mathbb Z}^{n-1}$action on ${\mathbb T}^n$ is C0 orbit equivalence rigid,as well as many other results of a similar nature. For the C0 category there are results when M1 is not a manifold, and we show that the orbit structure at infinity is a complete invariant for "most" hyperbolic groups. Unlike proofs of orbit equivalence rigidity in the measurable category, the proofs of our results are quite elementary and depend only on the countability of the group and the "size" of fixed point sets for group elements.

This is joint work with K. Whyte.

 

Eli Glasner (Tel-Aviv University)

Classifying dynamical systems by their recurrence properties

Abstract. In his seminal paper of 1967 on disjointness in topological dynamics and ergodic theory H. Furstenberg has shown that a dynamical system (X,T) is weakly mixing iff the collection ${\mathcal{F}}=\{N(U,V):U,V\subset X\ {\text{are non-empty open subsets}}\}$ is a filter base (here N(U,V) is the set of ``times" n such that $T^nU\cap V\ne\emptyset$). In recent years this fact served as a basis for a broad and detailed classification of topologically transitive dynamical systems by their recurrence properties. I will describe some aspects of this new and exciting theory and its connections with combinatorics, harmonic analysis and the theory of topological groups. Works by Glasner & Weiss (1993), Blanchard, Host & Maass (2000), Weiss (2000), Akin & Glasner (2001) and Huang & Ye (2002) will be reviewed.


Helmut Hofer (Courant Institute)

Holomorphic curves in low-dimensional dynamics

Abstract. The talk surveys some of the recent developments concerned with the use of holomorphic curves in the study of low-dimensional Hamiltonian systems. For example holomorphic curves can be used to construct global surfaces of section (or generalizations thereof) with quite minimal knowledge about the global dynamics. A complete picture can be obtained for large classes of Reeb flows on the three-sphere. The talk also surveys some of the open questions concerning the extension of the theory to autonomous Hamiltonian flows on three-dimensional energy surfaces of contact-type.


Suzanne Hruska (SUNY Stony Brook)

Constructing Hyperbolic Structures for Hénon maps

Abstract. We will describe a computer algorithm designed to prove hyperbolicity for polynomial diffeomorphisms of ${\ensuremath{\mathbb{R} } }^2$ or ${\ensuremath{\mathbb{C} } }^2$; in particular, for Hénon maps. We will give the classes of Hénon maps for which the computer-assisted proof has been successful, and discuss the numerical and dynamical obstructions found in attempting to prove hyperbolicity for certain other classes of maps.


Boris Kalinin (University of Michigan)

Rigidity properties of higher rank abelian actions

Abstract. We discuss rigidity of measurable structure for higher rank abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a 0 entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries. We also discuss some rigidity properties of nonalgebraic Anosov actions.


Yuri Kifer (Hebrew University)

Theorems and problems in fully coupled averaging

Abstract. Averaging set up arises in the study of systems obtained as a perturbation of systems having integrals of motion. This leads to a combination of a slow and a fast motion and the averaging prescription suggests to approximate the slow motion by the averaged one. When the fast motion does not depend on the slow one this prescription works, essentially, always and a lot is known about the averaging approximation and its error. In the general (fully coupled) case the situation is known to be more complex in view of resonances and the need to consider slowly changing dynamical systems. Some old and new results, as well, as remaining problems concerning this topic will be discussed.


Leonid Koralov (Princeton University)

Homogenization Transport by Random Flows

Abstract. We shall discuss several asymptotic problems for transport by stochastic flows. We prove the long time diffusive behavior for flows, which are Markovian in time. Next, for periodic stochastic flows we prove the Central Limit Theorem which holds with respect to the randomness in the initial conditions, for almost every realization of the flow. Finally we describe the limiting shape of a set carried by the flow. The latter results are joint with D. Dolgopyat and V. Kaloshin.


Rafael Krikorian (Ecole Polytechnique)

Differentiable rigidity and Lyapunov exponents: some examples

Abstract.


Patrice Le Calvez (Paris XIII)

A foliated equivariant version of Brouwer's Plane Translation theorem and some applications

Abstract. Let G be a discrete group of orientation preserving homeomorphisms acting freely on the plane. If f is an orientation preserving homeomorphism which commutes with the elements of the group and which is fixed point free., one may construct a topological foliation of the plane which is G-invariant and such that every leaf is a Brouwer line (it separates its image and preimage by f). This equivariant foliated version of the Brouwer plane translation theorem has some applications to the study of area preserving homeomorphisms of surface. We will present one of them: if M is a closed surface of genus $g\geq 1$ and F a hamiltonain homeomorphism (that means the time one map of a 1-periodic time dependent hamiltonian vector field), then F has an infinte number of periodic points corresponding to contractile periodic orbits of the vector field. The other ingedients in the proof are more classical objets of 2-dimentional dynamics: dynamics of foliations, Conley index for discrete maps, prime end theory, topological versions of the Poincaré-Birkhoff theorem.

 

Francois Ledrappier (Ecole Polytechnique)

Distribution results for cocompact groups in $SL(2,\mathbb{Q} _p)$ 

Abstract. This report on a joint work with Mark Pollicott. We continue our study of linear actions of lattices $\Gamma $ (and of their coabelian subgroups $\overline \Gamma $) in SL2 by investigating the case of the field of p-adic numbers. Results are similar to the ones in the cases of $\mathbb{R} $ or $\mathbb{C} $, with the following new features:

- an equidistribution result for the eigenvalues of matrices in $\Gamma $or $\overline \Gamma $.

- a complete description of the $\overline \Gamma $ invariant measures on the space ${\mathbb{Q} }_p \times {\mathbb{Q} }_p$.


Stefano Luzzatto (Imperial College)

Markov structures and decay of correlations for non-uniformly expanding dynamical systems

Abstract. We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure for which the decay of the return time function can be controlled in terms of the time generic points need to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem.


Stefano Marmi (University of Udine)

Quasianalytic solutions of cohomological equations

Abstract. (joint work with D. Sauzin)

Following a suggestion of Kolmogorov, Arnold and Herman studied the dependence on the multiplier of the solutions of cohomological equations in one complex dimension and in the analytic category (linearization of germs or of circle diffeos). Herman raised the question whether the solutions had a quasianalytic dependence on the multiplier. We show that the answer is affermative.


John Mather (Princeton University)

Arnold diffusion

Abstract. A form of Arnold's 1963 conjecture is true in the case of periodic Hamiltonians in two degrees of freedom and in the case of autonomous Hamiltonians in three degrees of freedom, provided that the unperturbed integrable Hamiltonian is convex.


Alica Miller (UIUC)

Characterizations of regular almost periodicity in compact minimal abelian flows (joint work with J. Rosenblatt)

Abstract. Regular almost periodicity in compact minimal abelian flows was characterized for the case of discrete acting group by W. Gottschalk and G. Hedlund and for the case of 0-dimensional phase space by W. Gottschalk a few decades ago. In 1995 J. Egawa gave characterizations of regular almost periodicity (of both, flow and a point in a flow) for the case when the acting group is $\mathbb{R} $. We extend Egawa's results to the case of an arbitrary abelian group and a not necessarily metrizable phase space. We then show how our statements imply previously known characterizations in each of the three special cases and give various other applications.


Kamlesh Parwani (Northwestern University)

Topologically monotone orbits on surfaces

Abstract. The notion of Birkhoff Orbits, as shown by Aubury-Mather theory, on the annulus for twist maps has been generalized to the idea of "Monotone Orbits" for more general homeomorphisms. We further generalize this to monotone periodic orbits on other surfaces and we obtain Sharkovski type theorems using the partial order as defined by Boyland.


Dan Rudolph (University of Maryland)

Relative mixing, isometric extensions and orbit equivalence

Abstract. For measure preserving and ergodic actions of $\mathbb Z$ the general understanding of isometric extensions is rather well understood. In particular one has a well-known list of results that all say if an action T has property A, and $\hat T$ is an isometric extension of T that is weakly mixing, then $\hat T$ must also have property A. Here one can put ``mixing", ``k-fold mixing", ``K", or ``Bernoulli" for property A. It has also been realized for many years that any result for an action should have a ``relativized" generalization, a version stated relative to an invariant factor algebra $\mathcal H$. Our goal is to consider the following question: Is it that case that an isometric extension of an action which is $\mathcal H$-relatively mixing, if it remains $\mathcal H$-relatively weakly mixing must in fact also be $\mathcal H$-relatively mixing?

A first step here is to understand what might be meant by $\mathcal H$-relatively mixing. Speaking vaguely, $\mathcal H$-relative mixing should mean that

 

\begin{displaymath}E(fg(T^n)\vert\mathcal H)- E(f\vert\mathcal H)E(g(T^n)\vert\mathcal H)\underset{n\to\infty}\rightarrow 0.\end{displaymath}

 

The question is, in what sense should this function be tending to zero. The natural answers are in L1 or pointwise. To see the difference in these, consider a [T,T-1]map where T is mixing and $\mathcal H$ is the base i.i.d. Bernoulli action. It is easy to see from the random walk that this action would be $\mathcal H$-relatively mixing if one asked for L1 convergence. But as the random walk is recurrent, it would be false for pointwise convergence.

What we will show is that the question above about isometric extensions can be answered in the affirmative for the pointwise definition. For the L1questions it remains open but the methods we use as we understand them will fail.

 

Victoria Sadovskaya (University of Michigan)

On local and global rigidity of uniformly quasiconformal Anosov systems

Abstract. We show that any uniformly quasiconformal contact Anosov flow on a compact manifold of dimension at least 5 is essentially smoothly conjugate to the geodesic flow of a manifold of constant negative curvature. In the discrete time case we show that any transitive uniformly quasiconformal Anosov diffeomorphism, whose stable and unstable distributions have dimensions greater than two, is smoothly conjugate to an Anosov automorphism of a torus. We also describe necessary and sufficient conditions for smoothness of conjugacy between such a diffeomorphism and its C1-small perturbation.


Martin Schmoll (Penn State)

On finite blocking properties for geodesics on translation surfaces

Abstract. One problem related to the asymptotic growth rate of various types of geodesic segments is to find $SL(2,\mathbb{R} )$ invariant subsets in moduli spaces of translation surfaces. We describe a way how to obtain $SL(2,\mathbb{R} )$invariant subsets for (branched) coverings of the flat torus by proving a finite blocking property, which roughly states that the every geodesic segment from the set of geodesic segments connecting two given points hits one of finitely many points (away from its endpoints). This simple fact for flat tori allows us to construct a lot of $SL(2,\mathbb{R} )$ invariant subsets for (branched) coverings of the flat torus. For torus coverings we give constructions, properties, some explanations and state questions related to the above ideas. As easy as results can be obtained for torus coverings as hard it seems to be, to make related statements for arbitrary translation surfaces.


John Smillie (Cornell University)

Complex dynamics in two dimensions

Abstract. I will discuss recent joint work with Eric Bedford and describe how this work fits in with the larger program of understanding two dimensional complex dynamics and the relations with two dimensional real dynamics.


Michael Sullivan (Southern Illinois University)

Flow equivalence of skew-products of irreducible shifts of finite type

Abstract. Let G be a finite group, and F be a function from the edge set of a given irreducible shift of finite type (SFT) X into G. This can be thought of as a skew-product system or a G-weighted SFT. A G-weighted SFT is determined by a square matrix over the semi-group ring Z+G. The following are known (W. Parry) to be equivalent conditions on two such matrices A and B arising from functions F and H, respectively,
(1) A and B determine isomorphic skew product systems,
(2) A and B are strong shift equivalence (SSE) over Z+G,
(3) there exists a topological conjugacy of their associated SFTs which sends F to a function cohomologous to H, and
(4) (if G is abelian) there exists a topological conjugacy of their associated SFTs which respects G-weights on periodic orbits.

A and B define flow equivalent skew products if their skew products can be made isomorphic after a time change. In the case G=Z/2Z, the equivalence relation is called twistwise flow equivalence, which has been applied to understand the twisting in the local stable manifolds of basic saddle sets in Smale flows. The twistwise flow equivalence classification has been open for five years.

Although complete written proofs have not yet been finished for checking, we think we have a proof of the following

Theorem. Let G be a finite group. Let A and B be square matrices over Z+Gdefining skew products of irreducible SFTs with G. Assume these SFTs are not trivial (i.e. contain more than one orbit). Then the skew products are flow equivalent if and only if the matrices I-A and I-Bare SL(ZG) equivalent, i.e., there exist matrices U, V over ZG with determinant 1 such that U(I-A)V = I-B.

This theorem grows out of the "positive K theory" framework in symbolic dynamics which has been applied in several categories. In fact the theorem is stronger: Any given SL equivalence (U,V) from I-A to I-B can be decomposed into a string of "positive" equivalences which yield flow equivalences.

Complete invariants for the algebraic relation of SL(ZG) equivalence are not hard to compute for G = Z/2 but the algebra becomes sophisticated even for G = Z/n. Expert algebraists know a lot about this (maybe everything).

We work in the setting of infinite, finitely supported matrices A(and the infinite identity I).

 

Andrew Torok (University of Houston)

Stable ergodicity of smooth compact Lie group extensions of hyperbolic basic sets

Abstract. We obtain sharp results for the genericity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over the basic set of a Cs diffeomorphism, $s \geq 2$. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the Cr topology for all $r\in(0,s]$ (except that C1 is replaced by Lipschitz).

Using our results we obtain stable transitivity for (non-compact) $\mathbb{R} ^m$-extensions over a general basic set, thereby generalizing a result of Nitica & Pollicott. We also obtain results on stability of weak mixing for hyperbolic suspension flows and Axiom A flows.

This is joint work with M. Field and I. Melbourne.


Ilie Ugarcovici (Penn State)

On admissible geometric codes for geodesics on modular surfaces

Abstract. Two different methods are available for coding geodesics on surfaces of constant negative curvature: a geometric one (Morse code) obtained by keeping track of the sides of a fundamental region hit by the geodesic, and an arithmetic one (Artin code) obtained by coding the endpoints of the geodesic (using continued fractions).

In this talk, we give a sufficient condition for a finite sequence of integers to be realizable as the geometric code of a closed geodesic on the modular surface. We will also discuss the problem for other modular surfaces, in particular for $\mathcal{H}/\Gamma(2)$, where $\mathcal H$ is the hyperbolic plane and $\Gamma(2)$ is the principal congruence subgroup of level 2.

This is joint work with Svetlana Katok.


Benjamin Weiss (Hebrew University)

Recent results in the theory of actions of amenable groups

Abstract. The last few years have seen remarkable progress in the extension of the classical ergodic theory from $\mathbb Z$ and $\mathbb{R} $ to amenable groups. I will give an overview of some of these, such as: Pointwise ergodic theorems, spectral and mixing properties of completely positive entropy actions, entropy estimation from finite data.


James Yorke (University of Maryland)

Learning about reality from observation

Abstract. (See www.math.umd.edu/$\sim$ott for a copy of the paper written with Will Ott).

2400 years ago Plato asked what we can learn from seeing only shadowy images of reality. In the 1930's Whitney studied "typical" images of manifolds in $\mathbb{R} ^m$ and asked when the image was homeomorphic to the original. Let A be a closed set in $\mathbb{R} ^{n}$ and let $\phi :\mathbb{R} ^{n} \to \mathbb{R} ^{m}$ be a ``typical'' smooth map where n > m. (Plato considered only the case n = 3, m = 2). Whitney's question has natural extensions. If $\phi (A)$ is a bounded set, can we conclude the same about A? When can we conclude the two sets have the same cardinality or the same dimension (for typical $\phi $)? (To simplify or clarify those questions, you might assume $\phi $ is a "typical" linear map in the sense of Lebesgue measure.)

In the 1980's Takens, Ruelle, Eckmann, Sano and Sawada extended this investigation to the typical images of attractors of dynamical systems. They asked when typical images are similar to the original. Now assume further that A is a compact invariant set for a map f on $\mathbb{R} ^{n}$. When can we say that A and $\phi (A)$ are similar, based only on knowledge of the images in $\mathbb{R} ^m$ of trajectories in A? For example, under what conditions on $\phi (A)$ (and the induced dynamics thereon) are A and $\phi (A)$homeomorphic? Are their Lyapunov exponents the same? Or, more precisely, which of their Lyapunov exponents are the same? This talk (and corresponding paper) addresses these questions with respect to both the general class of smooth mappings $\phi $ and the subclass of delay coordinate mappings.

 

Vadim Zharnitsky (Bell Labs)

Billiards and closed orbits in subriemannian geometry

Abstract. This is a joint work with Y. Baryshnikov. It is well known that there are exist non-circular billiards (corresponding to the equal-width curves) possessing a continuous family of 2-period orbits.

Using methods of subriemannian geometry, we prove analogous result for n-period orbits: There are exist non-elliptic billiards with a continuous family of n-period orbits.


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