Yakov Pesin - US grants

Professor Pesin will study the stochastic behavior of trajectories in smooth dissipative dynamical systems with singularities having attractors which are close to classical hyperbolic attractors. The object will be to describe the geometric structure of the attractors by calculating their dimension like characteristics. The principal investigator will also consider the appearance of space-time chaos in the lattice models which are constructed as a chain of strongly hyperbolic dynamical systems. This research involves random dynamical systems. These systems are used to model the behavior of large numbers of interacting particles and therefore the potential applications are significant. Previous work of Professor Pesin has recently become important because it gave considerable information on the dimension parameters of the attractors of the system. There is an indication that the current research will also find immediate application.

The proposed work involves a broad range of problems in the theory of dynamical systems (in particular, hyperbolic theory) and its applications to and relations with dimension theory, mathematical physics (including coupled map lattices), ergodic theory, and statistical mechanics. Projects in the dimension theory of dynamical systems include: the description of several multifractal spectra (including dimension spectra, entropy spectra, and spectra for Lyapunov exponents) and the corresponding multifractal decompositions for various classes of dynamical systems of hyperbolic type (non-conformal expanding maps, Axiom A diffeomorphisms, etc.); and the establishment of the multifractal rigidity phenomenon for multidimensional conformal expanding maps (the classification of dynamical systems up to multifractal spectra). In the theory of coupled map lattices (CML), the principal investigator will study: infinite-dimensional SRB measures for CML, which includes establishing the characteristic property and thermodynamical limit, describing the Lyapunov spectrum, and proving the entropy formula; and the stability of traveling wave solutions of CML and relations between traveling wave solutions for CML and the corresponding PDE (such as Ginzburg-Landau equation, Kolmogorov-Petrovski-Peskunov equation, Huxley equation, etc.). The principles of symmetry and self-similarity are nature's most beautiful creations. For example, they are expressed in fractals which are famous for their beautiful but complicated geometric shapes. Examples of fractals vary from well-known ones-cost lines or mountain ranges-to less known-distribution of stars in galaxies and galaxies in the universe or root systems of plants. Dimension theory is a mathematical theory which is designed to explain fractals' structure. And in dynamics the presence of invariant fractals often results in unstable ``turbulent-like'' motions and is associated with ``chaotic'' behavior. Thus the study of fractals can help understand the most complicat ed phenomena such as turbulence in the ocean or atmosphere. The proposed work involves research in a recently developing area which lies in the interface between dimension theory and the theory of dynamical systems. Focusing on invariant fractals and their influence on stochastic properties of systems, the principal investigator intends to provide a comprehensive and systematic treatment of modern dimension theory in dynamical systems. Results are expected to be of great importance not only to advanced mathematicians but to a wide range of scientists who depend upon mathematical modeling of dynamical processes, including physicists, specialists in numerical modeling, engineers, molecular biologists, etc.

This American Mathematical Society project supports a mathematical research institute during the summer of 1999. The institutes in this series bring together a group of mathematicians interested in a particular field of mathematical research. Emphasis is placed on instruction at a very high level with seminars and
lectures by distinguished mathematicians in related fields and to promote interaction between participants while broadening their mathematical perspectives.

Smooth Ergodic Theory and Applications has been selected as the topic of the 1999 Summer Research Institute by the AMS Committee on Summer Institutes and Special Symposia, whose members at the time were Michael D. Fried (chair), Robert Osserman, Jeffrey B. Rauch, Leon Takhtajan, and Ruth J. Williams.

Members of the Organizing Committee for the proposed 1999 Summer Research Institute on Smooth Ergodic Theory and Applications are Anatole Katok, Pennsylvania State University; Rafael De La Llave, University of Texas at Austin; Yakov Pesin, Pennsylvania State University; and Howard Weiss, Pennsylvania State University.

The 1999 Summer Research Institute will be held at the University of Washington, Seattle, Washington. The Society will be responsible for making suitable arrangements for lecture and seminar rooms, and for the
handling of local administrative matters. The Institute is expected to be held during a three-week period in late July and early August.

Abstract
Hu

This proposed research addresses ergodic properties of "almost hyperbolic" dynamical systems.
Here "almost hyperbolic" systems means a smooth dynamical systems that is hyperbolic everywhere except for a finite set of points. These systems include "almost Anosov" diffeomorphisms, "almost hyperbolic" invariant sets, piecewise expanding maps on the unit interval with indifferent fixed points, and invariant sets of expanding maps with indifferent fixed points. "Almost hyperbolic" systems lie on the boundary of the set of uniformly hyperbolic systems in the space of smooth dynamical systems. By the results obtained from "almost Anosov" diffeomorphisms on the surface, the ergodic properties of such systems may be quite different from that of uniformly hyperbolic systems, though the topological properties are similar. For example, these systems may or may not admit SRB measures, and even when they do, correlation decay may change from exponential to polynomial.
This project stresses existence and properties of SRB measures, rate of decay of correlations, and stochastic stability under small perturbations for general "almost hyperbolic" systems.

Uniformly hyperbolic systems are the main research subjects in smooth dynamical systems since late 60's. These systems display many types of complex dynamic behavior, and whose behavoir are often regarded as chaotic. Ergodic theory concerns understanding the long-term behavior of systems. Now ergodic properties for uniformly hyperbolic systems are understood very well, and people are interested in such properties for nonuniformly hyperbolic systems. "Almost hyperbolic" means that hyperbolic conditions fail at a finite set of points. These systems lie on the boundary of the set of uniformly hyperbolic systems, and are the simplest nonuniformly hyperbolic systems. Results obtained earlier for some particular systems, "almost Anosov" systems in the two-dimensional torus, show that such systems may-and sometimes do-exhibit totally different long term behavior. In this project we will study ergodic properties of more general "almost hyperbolic" systems. We are particular interested in orbit distrubitions, rate of mixing, and stochastic stability under small perturbation for such systems.

The proposed research deals with problems in the theory of smooth dynamical systems and their applications to mathematical and statistical physics and geometry. The main subject of study is the so- called hyperbolic dynamical systems that provides a mathematical foundation for the paradigm that is widely known as "deterministic chaos" -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This paradigm asserts that conclusions about global properties of a nonlinear dynamical system with sufficiently strong hyperbolic behavior can be deduced from studying the linearized systems along its trajectories. The study of hyperbolic phenomena originated in seminal works of Artin, Morse, Hedlund, and Hopf on the instability and ergodic properties of geodesic flows on compact surfaces. Later, hyperbolic behavior was observed in other situations (e,g, Smale horseshoes and hyperbolic toral automorphism). The systematic study of hyperbolic dynamical systems was initiated by Smale, Anosov and Sinai who studied dynamical systems with sufficiently strong hyperbolic behavior. Such systems possess high level of unpredictability and exhibit strong chaotic behavior. In the proposal the PI considers the weakest (hence, most general) form of hyperbolicity known as nonuniform hyperbolicity. The theory of nonuniformly hyperbolic dynamical systems originated in the work of the PI (sometimes this theory is referred to as "Pesin theory'') and the study of these systems is based upon the theory of Lyapunov exponents.

There are three main topics in the proposal. 1. Thermodynamic formalism for nonuniformly hyperbolic dynamical systems -- this is to build statistical physics of phase transitions for systems with nonzero Lyapunov exponents based on recent works on Markov extensions and tower constructions. 2. Mixed hyperbolicity and stable ergodicity -- this is to study how "typical" the systems with nonuniform hyperbolic behavior are. A recent result by Dolgopyat and the PI shows that such systems exist on any phase space. 3. Coexistence of hyperbolic and non-hyperbolic behavior -- this is to complement the famous Kolmogorov-Arnold-Moser (KAM) theory by constructing particular examples of systems with coexistence of nonzero Lyapunov exponents and areas with zero entropy. The PI also proposes to apply his work to the FitzHugh-Nagumo equation and the Brusselator model -- the famous models in neurobiology and chemistry. They provide interesting new and "naturally" appearing examples of nonuniformly hyperbolic systems as well as demonstrate transitions from relatively simple Morse-Smale systems to "strange" attractors and to Smale horseshoes.

0104675
Levi

This U.S. Mexico award will support Drs. Mark Levi, Howard Weiss, and Yakov Pesin in a research collaboration with Valentine Afraimovich of the Mathematics Department of the Unversidad Autonoma de San Luis de Potosi. The investigators plan to study: 1) Coupled map lattices corresponding to partial differential equations from physics and biology, in particular, the FitzHugh-Nagumo equation (which is of great interest in neurobiology). The collaborators intend to describe the ergodic properties of its local map and construct SRB measures for the attractor of this map. 2) The transition from coupled map lattices to partial differential equations via traveling waves, which will build a foundation for numerical modeling of some partial differential equations of evolution type. 3) The Dynamics of chains of coupled oscillators, in particular, those associated with the Sine-Gordon equation.

A coupled map lattice is a discrete time dynamical system whose phase space is of a particular form, and for which the overall system exhibits translational symmetry.
Coupled map lattices have recently gained wide popularity as models of spatio-temporal chaos and coherent structures. There is currently great interest in using coupled map lattices to model turbulence, nerve cells, phase transitions in statistical physics, and crystals. They also arise naturally from the discrete version of evolution partial differential equations.

This proposal is for the travel funds for American attendees of the
conference ``Kolmogorov and Contemporary Mathematics'', an International
scholarly meeting to commemorate the Centennial birthday of A. N.
Kolmogorov -- arguably one of the best mathematicians of the 20th
century. The conference will take place on June 16-21, 2003 in Moscow,
Russia and is expected to draw 600-800 participants from around the world.
Organized by the Russian Academy of Science and Moscow State University,
the Kolmogorov's Centennial Conference will be one of the largest and
most important gatherings of mathematicians in 2003.

The conference will cover many areas of mathematics and Plenary Talks
will be given by prominent mathematicians in these areas. It is
therefore, expected that the conference will have a great impact on many
areas of current active research in mathematics and its applications in
various areas of science.

This award will provide support for participants, especially women, graduate students, postdocs, and young faculty, in the "Workshop in Dynamical Systems and Related Topics." This is a long-standing series of conferences that are held each fall at Pennsylvania State University-University Park. The award will provide funding for the 2009, 2010, 2011, and 2012 conferences.

The theme of the Penn State dynamics workshop varies from year to year, but the primary focus areas are smooth ergodic theory, rigidity theory and actions of higher rank groups, and mechanics. The format of the conference provides ample opportunity for young mathematicians to present their work and otherwise to participate fully in conference activities.

This award provides support to defray expenses of US-based researchers participating in the international conference "Progress in Dynamics," which will be held at the Institut Henri Poincare in Paris from November 23-27, 2009. Most of the funding will be directed to graduate students, postdocs, and junior faculty, but certain senior participants may be funded as well.

The conference will cover a wide range of topics, touching upon virtually all the active areas in dynamical systems and ergodic theory. A unifying theme threading its way through the meeting's program is the impact of the work of Anatole Katok on the field. The format of the conference provides ample opportunity for young mathematicians to present their work and otherwise to participate fully in conference activities.

The "Penn State- Göttingen International Summer Schools in Mathematics" will be held during Summer 2010 at Penn State University, during Summer 2011 at the Georg-August-Universität in Göttingen, Germany, and during Summer 2012 at Penn State University. The summer school will be scheduled for three weeks and will have a different thematic focus each year. The first two weeks will concentrate on educational activities, while during the third week an international research conference will be held on the featured topic. The target audience for the project is undergraduate and beginning graduate students in mathematics who are based in the U.S. and in Germany. The summer schools integrate education and research and offer international experiences in instruction and collaboration for students at an early stage of their mathematical career. During each summer school, faculty from both Penn State and the University of Göttingen will conduct lecture courses and supervise student work. Students will work in international teams to accomplish theoretical work, work in a computer lab, and carry out comprehensive projects. The program will also facilitate networking between students and researchers. Students will experience the research atmosphere in the field at the international research conference. Part of the conference program will be an information forum about graduate work and graduate school application procedures and selection criteria.

The summer schools will contribute to attracting and retaining talented students to pursue graduate work and mathematical careers in academia, industry, and government, both in the United States and in Germany. They will provide international education and research experiences, promote the exchange of ideas and mutual understanding of different disciplinary cultures, and will enhance the ability of the participating students to engage in international collaborations in their future professional careers.

This grant is to provide travel support for American participants of two Workshops organized as part of a semester-long program "Hyperbolic Dynamics, Large Deviations and Fluctuations" that takes place at the Centre Interfacultaire Bernoulli, Ecole Politechnique, Lausanne, Switzerland, March-June 2013. The first workshop is on "Large deviations and thermodynamical formalism" and the second one is on "Limit Theorems for dynamical systems". Each workshop is preceded by a series of intensive mini-courses for post-graduate and post-doctoral students given by distinguished mathematicians in the field of dynamical systems.

During the last 50 years statistical properties of dynamical systems have been a center of extensive study in dynamics with many great applications to other areas of mathematics as well as outside to physics, engineering and biology. The systematic study of statistical properties began in the early 60's with the works of Sinai and Ruelle, and later of Bowen, who laid the foundations of thermodynamics. Though the methods they developed were of rather general nature, their applications were restricted mostly to uniformly hyperbolic systems. Much wider horizons were opened after the development of non-uniform hyperbolicity theory that is also known as "Pesin's theory". The main activities in dynamics these days are concentrated around studying non-uniformly hyperbolic systems, hyperbolic systems with singularities, and more generally, systems which posses some kind of hyperbolicity however do not belong to the previously mentioned categories. The main aim of the program is to bring together leading experts and young researchers working on statistical properties of dynamical systems and their applications in statistical mechanics, and mathematical physics as well as to foster scholarly exchange and collaboration. The workshops are intended to provide a focus of activities and a meeting place for mathematicians from various countries working in dynamical systems and related topics. They will serve as a forum for both young and established researchers to exchange ideas with their counterparts throughout the world, promote current and future research and create a concrete agenda for further research on several fronts. All participants, including members of traditionally underrepresented groups will also benefit from the presence of many strong young mathematicians from Europe.

This award will provide support for participants in the "Workshop in Dynamical Systems and Related Topics" for years 2013, 2014, 2015, and 2016. This is a long-standing series of conferences that are held each fall at Pennsylvania State University-University Park. The theme of the Penn State dynamics workshop varies from year to year; the focus areas include smooth ergodic theory, rigidity theory, smooth group actions, and classical mechanics.

The format of the conference provides ample opportunities for young mathematicians to present their work and otherwise to participate fully in conference activities. The bulk of the support goes to graduate students, postdocs, young faculty and women. A specific feature of the conference in the odd years is the announcement of the winner of the bi-yearly Michael Brin Prize in Dynamical Systems, endowed by the father of one the founders of Google, Professor Emeritus of the University of Maryland Michael Brin, and a special session featuring talks on the winner's work by the leading international experts in the area.

2020 Vision for Dynamics will be held in the Mathematical Research and Conference Center Bedlewo of the Institute of Mathematics at the Academy of Sciences of Poland from August 11 to 16, 2019. The conference will focus on topics related to elliptic, parabolic and hyperbolic dynamical systems and smooth ergodic theory that have seen much progress, but where significant problems vital to the field remain open. Based on forward-looking presentations of recent developments, the conference will set a broad and concrete agenda for further research on several fronts and bring together senior, mid-career and young practitioners for discussions of open problems in these research areas. More information can be found at: https://www.impan.pl/en/activities/banach-center/conferences/19-vision2020

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

This award provides participant support to a conference titled "Equidistribution, Invariant Measures and Applications", to be held on May 19 - May 24, 2019 at the Hebrew University of Jerusalem. The theory of actions of groups generated by unipotent elements also known as Ratner theory is a cornerstone of modern ergodic theory. The goal of the conference is to review the recent developments in the field and to set up new directions.

The main topics of the conference will be unipotent flows and their applications to counting and equidistribution; diagonal flows on homogenous spaces, and their applications in arithmetic and beyond; measure and orbit classification results for dynamics on moduli spaces of abelian and quadratic differentials; stationary measures and associated random walks in the homogeneous and non-homogeneous spaces. The conference structure will include mini-courses, lectures, and times for discussion and collaboration. More details about the conference could be found at http://ias.huji.ac.il/22midrashamathematicae .

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

This award provides funding for participants of “Semi-annual Workshop in Dynamical Systems and Related Topics at Penn State’' for the years 2022, 2023, and 2024, to be held at the Mathematics Department of Penn State University. The first of these meetings will take place on November 3-6, 2022. This represents the Penn State half of the Semi-annual Workshop in Dynamical Systems and Related Topics, co-organized for the last 32 years by the dynamics groups at Penn State and University of Maryland; the Maryland half is held each spring. The workshop is devoted to recent developments in dynamical systems and their numerous applications to related fields in mathematics such as algebra, geometry, and number theory.<br/> <br/>The goals of this conference are to promote the communication of mathematical results; to facilitate interaction and progress in the theory of dynamical systems viewed in a broad sense, and in related fields; to nurture the sense of community and common mission in these fields; and to contribute to the training of graduate students and recent Ph.D. recipients and to their integration into the dynamics community. Special efforts are made to attract women participants as well as members of underrepresented groups. The workshop is intended to provide a focus of activities and a meeting place primarily for mathematicians from the Northeast region of the United States working in dynamical systems and related topics, including the strong groups at Penn State and University of Maryland; and a forum for both young and established researchers from this region to help exchange ideas with each other and their counterparts from around the world. Conference information will be available at: https://science.psu.edu/math/research/dynsys/workshop<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.