PSU Mark
Eberly College of Science Mathematics Department

Abstract

Home » Dynsys » dw_archive » dw_2005 » Schedule
DYNAMICAL SYSTEMS and RELATED TOPICS WORKSHOP
October 14-17, 2005
Penn State

ABSTRACTS OF TALKS

 

 

Jose Alves (Universidade Porto, Portugal)

Topological structure of partially hyperbolic sets with positive volume.

Abstract. We show that partially hyperbolic sets with positive volume for diffeomorphisms whose class of differentiability is higher than 1 necessarily contain stable/unstable disks. In particular, there are no partially hyperbolic horseshoes with positive, thus generalizing a classical result by Bowen for uniformly hyperbolic horseshoes. We are also able to give a good description of the topological structure of (partially) hyperbolic sets with positive volume diffeomorphisms whose class of differentiability is higher than 1. This is a joint work with V. Pinheiro.

 

Michael Benediks (KTH, Stokholm, Sweden)

Non-uniformly hyperbolic attractors: invertible and non-invertible.

Abstract. In the talk the ergodic theory of non-uniformly hyperbolic attractors with positive Lyapunov exponent will be discussed -- in particular the case of attractors for non-invertible maps.

 

Pavel Bleher (IUPUI, Indianapolis)

Exact solution of the six-vertex model with domain wall boundary condition.

Abstract. The six-vertex model, or the square ice model, with domain wall boundary condition (DWBC) has been introduced and solved for finite N by Korepin and Isergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an NxN Hankel determinant. Paul Zinn-Justin observed that the Isergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large N asymptotics of the six-vertex model with DWBC in the disordered phase. The solution is based on the Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign matrices (the ASM problem) is a special case of the the six-vertex model. We compare the obtained exact solution of the six-vertex model with known exact results for the 1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free fermion line.

 

Leonid Bunimovich (Georgia Tech)

Dynamical systems and Benford's law.

Abstract. Benford's law (BL) is the empirical observation that in many numerical data the leading significant digits are not uniformly distributed but instead follow a particular logarithmic distribution. BL has been studied in Number theory and Probability theory. We will discuss when orbits of dynamical systems satisfy Benford's law. In particular, our approach allowed to recover all the results on BL in Number theory (and obtain various new examples) as well as explain why BL appears so often in real experimental data.

Jacek Graczyk (Univercity of Paris, Orsay, France)

Negative Schwarzian and cohomological inequality.

Abstract. We prove that analytic conjugacy class of $ C^{3}$ self map of a compact interval (circle) with all critical points non-flat and all periodic points repelling contains a map with negative Schwarzian derivative. The main idea is to solve a cohomological inequality in the class of essentially bounded measurable functions and then "smooth out" a solution. In case of circle maps we need one more condition to solve the cohomological inequality due to integral obstructions.

 

Yulij Ilyashenko (Cornell University and IUM, Moscow, Russia)

Skew products and nonremovable Lyapunov exponents.

Abstract. Recently Gorodetski, Kleptsyn, Nalski and the speaker found an open set in the space of dynamical systems on a three-torus, for any map of which there exists a nonhyperbolic nonatomic ergodic invariant measure. Nonhyperbolicity means that the measure has an (intermediate) zero Lyapunov exponent. This result is a part of a large program suggested by Gorodetski and Ilyashenko in 1998.

 

Francois Ledrappier (University of Notre Dame)

Distributions of horocycles on abelian covers.

Abstract. We consider the horocycle flow $ h_s$ on an abelian $ d$ -dimensional cover of a compact hyperbolic surface. The invariant ergodic Radon measures form a continuous family indexed by $ {\mathbb{R}}^d$ . We show that out of them, only the Lebesgue measure $ m_0$ is rationally ergodic. Moreover, the Lebesgue measure satisfies the following ergodic theorem: for every $ f\in L^1$ , $ m_0$ -almost every $ \omega $ :

$\displaystyle \lim _{N \to \infty } \frac {1}{\ln \ln N} \int _3^\infty \frac {... ...int _0^T f(h_s\omega ) ds \right) \frac {dT}{T\ln T}\quad = \quad \int f dm_0, $

 

where

$ a(T) = (\ln T)^{d/2} /T $ and $ C$ is an explicit constant. This is a joint work with O . Sarig.

 

Stefano Luzzatto (Imperial College, London, UK)

25 years on: a quantitative, computer-assisted, version of Jakobson's Theorem.

Abstract. We formulate and prove a Jakobson-Benedicks-Carleson type theorem on the occurrence of non-uniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters.

As an application of our results we obtain a first ever explicit lower bound for the measure of the set of parameters corresponding to maps in the quadratic family $ f_{a}(x) = x^{2}-a $ which have an absolutely continuous invariant probability measure.

This is joint work with H. Takahasi (Kyoto University).

 

Gregory Margulis (Yale University)

Discrete groups of affine transformations.

Abstract. I will discuss old and recent results about proper affine actions of discrete groups on $ R^n$ . The emphasis will be on the Auslander conjecture and on the study of the set of proper affine actions whose linear part is a Schottky group.

 

Michal Misiurewicz (IUPUI, Indianapolis)

Invariant measures for random iterations of affine maps.

Abstract. We consider actions of the free semigroup with 2 generators on the real line. The generators act as affine maps, one contracting and one expanding, with distinct fixed points. Then every orbit is dense in a half-line, and we can ask whether it is, in some sense, uniformly distributed. We present answers to this question for various interpretations of the phrase "uniformly distributed". The results were obtained jointly with Vitaly Bergelson and Samuel Senti.

 

Sheldon Newhouse (Michigan State University)

Dissipative surface dynamics: critical points vs. induced maps.

Abstract. We compare two fundamentally different approaches to prove the existence of SRB measures in parametrized area decreasing surface diffeomorphisms. The first, pioneered by Benedicks-Carleson and Benedicks-Young for the Henon family, and subsequently extended by Mora-Viana, Luzatto-Viana, and Wang-Young, seeks to find "good" parameters for which one can define a "critical set" whose forward orbits determine the dynamics. The second, which is part of on-going work of Michael Jakobson and myself, seeks to find "good" parameters where there is a certain power or induced map which is uniformly hyperbolic. Thus, in a certain sense, the critical points have been eliminated from consideration. This latter approach is inspired by Jakobson's original proof of the abundance of absolutely continuous invariant measures for quadratic interval maps.

 

Samuel Senti (IMPA, Rio de Janeiro, Brazil)

Measures of maximal entropy for a class of non-uniformly hyperbolic 2-dimensional maps.

Abstract. We study perturbations of the skew-product on $ S^1\times\mathbb{R}$ given by

$\displaystyle f(x, y)=(16x, a+\epsilon\sin(2\pi x)-y^2)$

 

for some small

$ \epsilon>0$ and a parameter $ a$ for which $ y=0$ is preperiodic for the map $ a-y^2$ . We show how to prove the existence of equilibrium measures for potentials with small variations including constant potentials.

 

Sebastian van Strien (University of Warwack, UK)

Solution of the real Fatou conjecture: from Jakobson to the present day.

Abstract. We present a solution of the real Fatou conjecture, which states density of hyperbolicty for maps of the interval or the circle. We also will discuss some consequences of this, in particular monotonicity of entropy.

 

Greg Swiatek (Penn State University)

Dynamics of meromorphic Misiurewicz maps.

Abstract. Properties of meromorphic transcendental maps of the Riemann sphere will be discussed. Several key theorems known for rational maps still hold in this case.

However, an important difference exists in the criterion for the existence of a probabilistic absolutely continuous invariant measure, which in the transcendental case requires an additional hypothesis about the essential singularity at infinity. This condition can be expressed in terms of the Nevanlinna characteristic of the function. (joint work with J. Kotus)

 

Masato Tsujii (Hokkaido University, Japan)

Decay of correlations in expanding semi-flows.

Abstract. I discuss about decay of correlations in suspension semi-flows of angle-multiplying maps on the circle. It has been shown by M. Pollicott (based on Dolgopyat's argment) that, under a mild condition on the ceiling function, the decay rate is exponential. In this talk, I present the following result: under a generic condition on the ceiling function, the Perron-Frobenius operator acting on the anisotropic Sobolev space introduced by V. Baladi and the author has essential spectral radius not greater than the square root of the inverse of the minimal expansion rate. This leads to a precise asymptotic expansion of the correlations.

 

Marcelo Viana (IMPA, Rio de Janeiro, Brazil)

TBA

Abstract.

 

Lai-Sang Young i(NYU, Courant Institute)

Detecting strange attractors.

Abstract. First I will describe the type of attractors I hope to detect, namely those with a single direction of instability and strong contraction in all other directions. Three dynamical situations geometrically compatible with the presence of these attractors will be discussed, and conditions that guarantee their existence will be given.

 

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