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

Introduction to the Modern Theory of Dynamical Systems

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

Let X be a compact real analytic manifold, and let T* X be its cotangent bundle. In a recent paper with Zaslow (J Am Math Soc 22:233–286, 2009), we showed that the dg category Sh
 c
 (X) of constructible sheaves on X quasi-embeds into the triangulated envelope F(T* X) of the Fukaya category of T* X. We prove here that the quasi-embedding is in fact a quasi-equivalence. When X is a complex manifold, one may interpret this as a topological analogue of the identification of Lagrangian branes in T* X and regular holonomic $${{\mathcal D}_X}$$
 -modules developed by Kapustin (A-branes and noncommutative geometry, arXiv:hep-th/0502212) and Kapustin and Witten (Commun Number Theory Phys 1(1):1–236, 2007) from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians in T* X (with some modest homological assumptions) are equivalent in the Fukaya category to the zero section. In particular, this determines their (complex) cohomology ring and homology class in T* X, and provides a homological bound on their number of intersection points. An independent characterization of compact branes in T* X has recently been obtained by Fukaya et al. (Invent Math 172(1):1–27, 2008).

Microlocal branes are constructible sheaves

We discuss various algebraic quantum structures associated to monotone Lagrangian submanifolds and we present a number of applications, computations and examples.

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Quantum Structures for Lagrangian Submanifolds

This paper explores the topology of monotone Lagrangian submanifolds L inside a symplectic manifold M by exploiting the relationships between the quantum homology of M and various quantum structures associated to the Lagrangian L.

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Rigidity and uniruling for Lagrangian submanifolds

We use Floer cohomology to prove the monotone version of a conjecture of Audin: the minimal Maslov number of a monotone Lagrangian torus in ℝ2n is 2. Our approach is based on the study of the quantum cup product on Floer cohomology and in particular the behavior of Oh's spectral sequence with respect to this product. As further applications, we prove existence of holomorphic disks with boundaries on Lagrangians as well as new results on Lagrangian intersections.

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The Maslov class of Lagrangian tori and quantum products in Floer cohomology

We introduce new invariants associated with collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants involves various flavors of Floer theory, including the μ
 3-operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics.

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Poisson brackets and symplectic invariants

Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms. 
Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point, shows that the conjecture does not admit a direct generalization to the C^0 setting. However, in this paper we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.

The action spectrum and C^0 symplectic topology

In this paper we find homological restrictions on Lagrangians in cotangent bundles of spheres and Lens spaces.

Homology of Lagrangian submanifolds in cotangent bundles

In this paper we introduce a new method for approaching the C
 0-rigidity results for the Poisson bracket. Using this method, we provide a different proof for the lower semi-continuity under C
 0 perturbations, for the uniform norm of the Poisson bracket. We find the precise rate for the modulus of the semi-continuity. This extends the previous results of Cardin–Viterbo, Zapolsky, Entov and Polterovich. Using our method, we prove a C
 0-rigidity result in the spirit of the work of Humiliere. We also discuss a general question of the C
 0-rigidity for multilinear differential operators.

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Lev Buhovsky

Papers

The Maslov class of Lagrangian tori and quantum products in Floer cohomology

Poisson brackets and symplectic invariants

The action spectrum and C^0 symplectic topology

Homology of Lagrangian submanifolds in cotangent bundles

The 2/3-Convergence Rate for the Poisson Bracket