/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

Tables of integrals

Mathematica--A System for Doing Mathematics by Computer.

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

Functions of One Complex Variable

Spectral geometry of the Steklov problem (Survey article)

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. While this problem shares some common properties with its more familiar Dirichlet and Neumann cousins, its eigenval- ues and eigenfunctions have a number of distinctive geometric features, which makes the subject especially appealing. In this survey we discuss some recent advances and open questions, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry.

/pdf/spectral-geometry-of-the-steklov-problem-4buu5i2m06.pdf

Spectral geometry of the steklov problem

We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of a twice smaller area. This estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Polya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.

/pdf/maximization-of-the-second-positive-neumann-eigenvalue-for-3vqizb0mdd.pdf

Maximization of the second positive Neumann eigenvalue for planar domains

Vladimir Andreevich Steklov, an outstanding Russian mathematician whose 150th anniversary is celebrated this year, played an important role in the history of mathematics. Largely due to Steklov’s efforts, the Russian mathematical school that gave the world such giants as N. Lobachevsky, P. Chebyshev, and A. Lyapunov, survived the revolution and continued to flourish despite political hardships. Steklov was the driving force behind the creation of the Physical–Mathematical Institute in starving Petrograd in 1921, while the civil war was still raging in the newly Soviet Russia. This institute was the predecessor of the now famous mathematical institutes in Moscow and St. Petersburg bearing Steklov’s name.

/pdf/the-legacy-of-vladimir-andreevich-steklov-1p3sfzczz8.pdf

The legacy of Vladimir Andreevich Steklov

The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a sphere S 4 by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawson's �3,1-torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for �1 on a Klein bottleofagivenarea. Wepresentnumericalevidenceandprovethefirstresultstowardsthisconjecture.

/pdf/extremal-metric-for-the-first-eigenvalue-on-a-klein-bottle-45r6umntov.pdf

Iosif Polterovich

Papers

Spectral geometry of the Steklov problem (Survey article)

Spectral geometry of the steklov problem

Maximization of the second positive Neumann eigenvalue for planar domains

The legacy of Vladimir Andreevich Steklov

Extremal Metric for the First Eigenvalue on a Klein Bottle