MD-PS 2001 Talks Titles and Abstacts

# TITLES AND ABSTRACTS OF TALKS

## Maryland-Penn State Semiannual Workshop on Dynamical Systems and Related Topics University of Maryland, March 17-20, 2001

Karen Ball, University of Maryland
Skew products which are not standard
Abstract form: dvi or pdf.

Henk Bruin, University of Groningen
Expansion of derivatives and the measure of Julia sets
This is a report on joint work with Sebastian van Strien (University of Warwick). Given a rational map on the Riemann sphere, we show that summable growth rates of the derivatives along the critical orbits imply similar growth rates for every non-precritical orbit in the Julia set. This is used to prove that a rational map satisfying such condition has a Julia set which is either the whole sphere, or has Lebesgue measure zero.

William Cowieson, New York University
Stochastic stability for invariant densities of piecewise expanding maps
Under suitable regularity conditions, "most" piecewise expanding maps T of a d-dimensional domain have an invariant measure which is absolutely continuous w.r.t. Lebesgue measure. Moreover, this holds also for suitable random perturbations of T, and the stationary distributions obtained tend to invariant densities of T as the noise tends to 0.

Beverly Diamond, College of Charleston
A complete invariant for the topology of 1-d substitution tiling spaces
This talk is based on joint work with Marcy Barge (Montana State University). Given a primitive, nonperiodic substitution $\varphi$, the associated tiling space ${\mathcal T}_\varphi$ has a finite (nonzero) number of asymptotic composants. We describe the form of these asymptotic composants and give a bound for their number. We make use of this description to discuss the topology of tiling spaces and provide examples indicating that for substitution minimal systems, flow equivalence and orbit equivalence are independent.

Filiz Dogru, Pennsylvania State University
On polygonal dual billiards in the hyperbolic plane
The dynamics of the polygonal dual billiard map depends on the size and shape of the respective polygon. For a class of polygons, called "large", all orbits of the dual billiard map escape to infinity (in sharp contrast with the Euclidean case).

Dmitry Dolgopyat, Penn State University
Differentiation of SRB measures
We discuss how invariant measures, Invariant plane fields, Lyapunov exponents etc. of dynamical systems depend on parameters in the class of partially hyperbolic sistems.

John Franks, Northwestern University
Group actions on one-manifolds
(Mathematics Department Colloquium Talk, Friday March 16, 3 p.m.)

Kresimir Josic, Boston University
The Structure of the Julia Set of Stable Exponentials
We consider the dynamics of the complex exponential $\lambda e^z$, where $\lambda$ is chosen so that the dynamical system defined by the map has an attracting periodic orbit. If the attracting orbit consists of a fixed point, the Julia set for this map is a Cantor bouquet and it has been well studied. We show that in the case the period of the attracting orbit is greater than one a set similar to a Cantor bouquet still exists, but some of the hairs comprising the bouquet are pinched together. The structure of this set and the accessibility of the points within it will be discussed.

Dynamics of an Oil Spot on the Surface of Ocean and Dynamics with Nonzero Lyapunov Exponents
Consider a surface of ocean (2-dimensional pane) and shade a unit disk in the origin (an oil spot or a passive scalar). Evolition of the surface of ocean in time is described by an area preserving Stochastic Differential Equation. What is an assymptotic behavour of this Oil Spot? What is its diameter for large time? and so on. It turns out that different features of evolution of such a spot generically can be described using methods of systems with nonzero Lyapunov exponents. This will be report on a joint work with Dima Dolgopyat and Lenya Koralov.

Bryna Kra, Penn State University
Some non-conventional ergodic averages
I will discuss recent results on non-conventional ergodic averages, including products of functions and IP-like expressions. For such averages, the limiting function is, in general, not constant but is determined by a factor. This is joint work with Bernard Host.

Francois Ledrappier, CNRS and Northwestern University
A weak Besicovich property and applications
The Besicovich property is a strong tool of geometric measure theory. When computing or estimating Hausdorff dimensions, the full strength of the property is not needed. We propose a weaker property and present (mostly future) applications.

Elon Lindenstrauss, Institute for Advanced Study
I will consider a very simple dynamical system, namely the 1-torus $R/Z$ with the map $x \mapsto mx mod 1$, and discuss some long-standing conjectures regarding this map. I will also show how one can use $p$-adic analogues of results from the theory of multidimensional smooth dynamical systems in this context, and in particular, we show that if $\mu$ is ergodic under the times $m$ map and has positive entropy, then $\mu$-a.s. $\{r ^n x\}$ is equidistributed as long as $m$ does not divide any power of $r$ (this was previously known only if $r$ and $m$ are relatively prime).

Brian Martensen, Montana State University
The local topological complexity of C^r-diffeomorphisms near homoclinic tangency
If F is a C^r-diffeomorphism with periodic saddle p so that a branch of the unstable manifold of p has a tangency with the stable manifold, then C^r-close to F is a diffeomorphism such that the closure of this branch of the unstable manifold everywhere locally contains all chainable continua.

Alica Miller, Michigan State University
Minimality of restrictions, group-extensions and products of compact minimal abelian flows
Abstract form: dvi or pdf.

Linda Moniz, University of Maryland
Upper Bound Sets for Dynamical Behavior
We describe an upper bound method for finding periodic points and other relevant features of $C^2$ maps in the plane. This method produces a set D that provides necessary conditions for dynamical behavior, i.e. if a point exhibits the behavior (e.g. a periodic point of period k) it must be in the set D. In many cases we can find a nested sequence of sets D that converges to the set of points with the desired property.

Nicholas S. Ormes, University of Connecticut
Symbolic Dynamics and Ordered Cohomology.
The speaker will discuss the use of ordered cohomology groups as invariants for symbolic dynamical systems. Various authors (e.g. Boyle/Handelman, Giordano/Putnam/Skau) have used the order structure to form a stronger invariant than the cohomology groups alone. More recent results (e.g. Ormes/Radin/Sadun, Yi) seem to suggest that the invariance of certain ordered cohomology groups is a purely topological phenomenon, not relying on underlying dynamics.

Dhanurjay (DJ) Patil University of Maryland
Local low dimensionality of atmospheric dynamics -- when good forecasts go bad
From the dynamical systems point of view, the behavior of the Earth's atmosphere is extremely high dimensional (e.g., a realistic atmospheric model based on a modal expansion would necessarily include many modes). In spite of the atmosphere's high dimensionality, in this talk I will demonstrate that, in a suitable sense, the local finite-time atmospheric dynamics is often low dimensional. Furthermore, I will show how this finding has important implications for weather forecasting. More generally, this behavior may be common to other physical spatio-temporally chaotic systems, and these systems may also be amenable to the type of analysis that is introduced for the atmosphere.

Yakov Pesin, Penn State University
On the Existence of Bernoulli Diffeomorphisms With Nonzero Lyapunov Exponents
We show that a smooth compact Riemannian manifold of dimension $\ge 2$ admits a Bernoulli diffeomorphism with nonzero Lyapunov exponents.

Karl Petersen, University of North Carolina
From K to Super-K
The future tail field of a finite-state stationary process consists of all the information that remains after time has run its course; for Kolmogorov processes, this tail field is trivial. The future fine tail field consists of the residual information in case a limited amount of record keeping, namely symbol counts accumulated from time 0 onward, is allowed; for Bernoulli processes the future fine tail field is also trivial, but it need not be trivial for other partitions that generate the process. With J.-P. Thouvenot, we have been working to recode Kolmogorov processes in such a way that the future fine tail field is trivial. We are able to accomplish such a recoding in case there is a direct Bernoulli factor by using the asymptotic flatness of the distribution of symbol counts in Bernoulli processes.

Mason Porter, Cornell University
Quantum Chaos in Vibrating Billiard Systems
We consider quantum billiards with vibrating surfaces, which can be used as a semiquantal description of nonadiabatic coupling in polyatomic molecules and nanomechanical devices. We review the various notions of quantum chaos, focusing on that observed in semiquantal systems. We derive necessary conditions for chaotic behavior in quantum billiards with a single degree-of-vibration, and we discuss the relationship between the observed quantum chaos (which is known as 'semiquantum chaos') and classical Hamiltonian chaos. The present analysis is illustrated by considering the radially vibrating spherical quantum billiard.

Anthony Quas, University of Memphis
Convexity, Random Tilings and Shifts of Finite type
In one dimension, it is known that a mixing shift of finite type has a unique measure of maximal entropy and this measure is isomorphic to a Bernoulli shift. This fails badly in higher dimensions. In this talk, we find a class of higher-dimensional shifts of finite type for which there are Bernoulli measures with entropy arbitrarily close to the topological entropy - this can be viewed as a strong variational principle for these systems. On the way, we establish some general facts about higher-dimensional entropy.

Rafael Oswaldo Ruggiero, Pontificia Universidade Catolica do Rio de Janeiro
On the divergence of geodesic rays in manifolds without conjugate points
Let $(M,g)$ be a compact Riemannian manifold without conjugate points. Denote by $H(p,v)$ the horosphere containing $p$, in the universal covering $\tilde{M}$, of the geodesic $\gamma_{(p,v)}(t)$ (defined by $\gamma_{(p,v)}(0)=p$, $\gamma_{(p,v)}'(0)=v$). Suposse that the map $$(p,v) \longrightarrow H(p,v)$$ is continuous (uniformly on compact subsets of $\tilde{M}$). Then we prove that the distance $d(\gamma(t),\beta(t))$ between two different geodesic rays starting at any point $q \in \tilde{M}$ tends to infinity with $t$. We use this fact to show that a continuous local product structure of horospheres in $T_{1}M$ implies that the geodesic flow is expansive, and hence $\tilde{M}$ is a Gromov hyperbolic space.
This result can be viewed as a topological counterpart of a wellknown theorem of P. Eberlein which says that the Anosov property is equivalent to the transversality of Green bundles.

Martin Sambarino, University of Maryland
On the existence of homoclinic orbits
We will show that there exists an open and dense set ${\cal{U}}$ in $Diff^1(M)$, where $M$ is any compact manifold, such that each $f\in{\cal{U}}$ either is a Morse-Smale diffeomorphism or exhibits a transverse homoclinic orbit.

Lai-Sang Young, New York University
From invariant curves to strange attractors
Simple mechanical systems, when subjected to external periodic forcing, can exhibit a rich array of dynamical behaviors as parameters are varied. I will discuss among other things the existence of global strange attractors with fully stochastic properties for a class of second order ODEs.

Michiko Yuri, Sapporo University
Weak Gibbs measures for potentials of weak bounded variation and subexponential instability
Abstract form: dvi or pdf.

Piotr Zgliczynski, Indiana University, Bloomington
Isolating segments and chaotic behavior.
Using topological tools (isolating segments) we prove that the differential equation on the plane $z'=\bar{z}(1+|z|^2 e^{i\kappa t })$ has a symbolic dynamics on three symbols for $0 \kappa \leq 0.495$. We show also an existence of an infinite number of geometrically distinct solutions, which are homoclinic to the constant zero solution.