A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Topological methods in hydrodynamics

There is a mysterious relation between rigidity phenomena of symplectic geometry and global periodic solutions of Hamiltonian dynamics. One of the links is provided by a special class of symplectic invariants discovered by I. Ekeland and H. Hofer in [2], [3] called symplectic capacities. We first recall this concept in a more general setting from [26] and consider the class of all symplectic manifolds (M, ω) possibly with boundary, but of fixed dimension 2n. Here ω is a symplectic structure, i.e. a two-form on M which is closed and nondegenerate.

Symplectic Invariants and Hamiltonian Dynamics

Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the formWe assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.

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On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN

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

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

Introduction Part 1. Cobordism: Hamiltonian cobordism Abstract moment maps The linearization theorem Reduction and applications Part 2. Quantization: Geometric quantization The quantum version of the linearization theorem Quantization commutes with reduction Part 3. Appendices: Signs and normalization conventions Proper actions of Lie groups Equivariant cohomology Stable complex and Spin$^{\mathrm{c}}$structures Assignments and abstract moment maps Assignment cohomology Non-degenerate abstract moment maps Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization The Kawasaki Riemann-Roch formula Cobordism invariance of the index of a transversally elliptic operator Bibliography Index.

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Moment Maps, Cobordisms, and Hamiltonian Group Actions

Poisson Lie groups appeared in the work of Drinfel'd (see, e.g., [Drl, Dr2]) as classical objects corresponding to quantum groups. Going in the other direction, we may say that a Poisson Lie group is a group of symmetries of a phase space that are allowed to "twist," in a certain sense, the symplectic or Poisson structure. The Poisson structure on the group controls this twisting in a precise way. Quantizing both the phase space and the symmetries, one may obtain a quantum group acting on a quantum phase space. In recent work, Lu and Ratiu [LR] used so-called standard Poisson structures on a compact semisimple Lie group K and on its Poisson dual K* in order to give a new proof of the nonlinear convexity theorem of Kostant [Ko]. Their method is analogous to the famous symplectic proof of the linear convexity theorem given by Atiyah [A] and Guillemin and Stemnberg [GS]. The nonlinear convexity theorem, like the linear one, follows from a very general result on convexity of the image of the momentum map [A, GS]; however, in [LR] it is applied not to a coadjoint orbit in t*, but to a symplectic leaf in K* . The main result of this paper is that the standard Poisson structure on the Poisson dual K* to a compact semisimple Poisson Lie group K is actually isomorphic to the linear one on t* . This theorem seems to be related to some facts in the theory of quantum groups. Namely, (for generic q) the universal enveloping algebra U(t) and its quantum deformation Uq(t) are isomorphic as algebras (though not as coalgebras, of course, since the quantum version is not cocommutative). In particular, there is a bijective correspondence between their representations. A direct connection between our work and its quantum analogues, though, is still to be found. The present work supplies a positive answer to Question 5.1 in [LR] and strongly depends on that paper. To simplify reading, we keep the notations of [LR] wherever possible, on one hand, but give all necessary definitions, on the other. Our work is also related to [Du], in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in ?5. The paper is organized as follows. In ?2 we define Poisson Lie groups, discuss

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Lie-Poisson structure on some Poisson Lie groups

We prove the Conley conjecture for a closed symplectically aspherical symplectic manifold: a Hamiltonian diffeomorphism of such a manifold has infinitely many periodic points. More precisely, we show that a Hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period. This theorem generalizes, for instance, a recent result of Hingston establishing the Conley conjecture for tori.

http://annals.math.princeton.edu/wp-content/uploads/annals-v172-n2-p07-p.pdf

The Conley Conjecture

In this paper, we study the behavior of the local Floer homology of an isolated fixed point and the growth of the action gap under iterations. We prove that an isolated fixed point of a diffeomorphism remains isolated for the so-called admissible iterations and that the local Floer homology groups of a Hamiltonian diffeomorphism for such iterations are isomorphic to each other up to a shift of degree. Furthermore, we study the pair-of-pants product in local Floer homology, and characterize a particular class of isolated fixed points (the symplectically degenerate maxima), which plays an important role in the proof of the Conley conjecture. Finally, we apply these results to show that for a quasi-arithmetic sequence of admissible iterations of a Hamiltonian diffeomorphism with isolated fixed points the minimal action gap is bounded from above when the ambient manifold is closed and symplectically aspherical. This theorem is a generalization of the Conley conjecture.

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Local Floer homology and the action gap

The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We give examples of functions with a sequence of regular levels without periodic orbits, converging to an isolated, but very degenerate, minimum. The proof of the relative almost existence theorem hinges on the notion of the relative Hofer-Zehnder capacity and on showing that this capacity of a small neighborhood of a symplectic submanifold is finite. The latter is carried out by proving that the flow of a Hamiltonian with sufficiently large variation has a nontrivial contractible one-periodic orbit when the Hamiltonian is constant and equal to its maximum near a symplectic submanifold and supported in a neighborhood of the submanifold.

Viktor L. Ginzburg

Papers

Moment Maps, Cobordisms, and Hamiltonian Group Actions

Lie-Poisson structure on some Poisson Lie groups

The Conley Conjecture

Local Floer homology and the action gap

Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles