Symplectic invariants and Hamiltonian dynamics.

*(English)*Zbl 0805.58003
Birkhäuser Advanced Texts. Basel: Birkhäuser. xiii, 341 p. (1994).

Symplectic Topology has become a fascinating subject of research over the past fifteen years. It is astonishing to see how Analysis, Differential Geometry, Algebraic Topology and Physics interplay in such a beautiful way to form the body of the subject to which this book is devoted.

The origin of this theory can be traced back to Poincaré’s geometrical theorem (proved by Birkhoff): “Any area preserving map of the annulus on itself such that it moves its boundary circles on opposite directions, has at least two fixed points.”

But it was not until the advent of Arnol’d’s conjecture [V. I. Arnol’d, Russ. Math. Sur. 41, No. 6, 1-21 (1986); translation from Usp. Mat. Nauk 41, No. 6, 3-18 (1986; Zbl 0618.58021)] that symplectic topology has emerged as a subject of its own: If \(\phi\) is a nondegenerate exact symplectic diffeomorphism on a compact symplectic manifold, then \(\phi\) has at least as many fixed points as the Morse number of the manifold. Chapter 6 of the book under review deals with this conjecture within the spirit of Floer homology.

Reformulated dynamically, this conjecture gives a lower bound for the number of global forced oscillations of a time-dependent Hamiltonian vector field on a compact symplectic manifold in terms of the topology of this manifold, or in terms of the number of critical points for smooth functions on this manifold. The conjecture has been solved only for certain cases. For example, C. Conley and the second author [Invent. Math. 73, 33-49 (1983; Zbl 0516.58017)] proved it for the torus, and B. Fortune [Invent. Math. 81, 29-46 (1985; Zbl 0547.58015)] for the complex projective space. Later on, A. Floer [Duke Math. J. 53, 1-32 (1986; Zbl 0607.58016)] was able to prove the conjecture for symplectic manifolds with vanishing second homotopy group, and later on, for monotone symplectic manifolds [same author, J. Differ. Geom. 30, No. 1, 207-221 (1989; Zbl 0678.58012)]. Floer’s ideas are based on a fresh use of Morse theory as rephrased by E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)].

A combination of Floer’s construction and the special capacity \(c_ 0\) of chapter 3, allows the authors to present a symplectic homology theory. But this chapter is, in contrast to the previous chapters, not self- contained.

However, Arnol’d’s conjecture is not the central point of the book. The main theme is a class of symplectic invariants; the so-called symplectic capacities.

A symplectic capacity is a map \((M,\omega)\to c(M,\omega)\), where \((M,\omega)\) is a symplectic manifold possibly with boundary and of fixed dimension \(2n\), which assigns to each \((M,\omega)\) a nonnegative real number or \(\infty\) satisfying monotonicity, conformality and nontriviality. The main difference with volume symplectic invariants is that \(c(Z(1),\omega_ 0) < \infty\), where \(Z(1)\) is the open symplectic cylinder in the standard space. Chapter 2 is devoted to the axiomatic definition of symplectic capacities and to their application to embeddings. For example, Gromov’s well-known squeezing theorem is a consequence of the existence of symplectic capacities.

Chapter 3 outlines the existence proof of a distinguished capacity function, \(c_ 0\), that was introduced by the authors in Analysis et cetera, Res. Pap. in Honor of J. Moser’s 60th Birthd., 405-427 (1990; Zbl 0702.58021). This capacity measures the minimal \(C^ 0\)-oscillation of special Hamiltonian functions needed in order to conclude the existence of a distinguished periodic orbit having small period and solving the associated Hamiltonian system. The special features of \(c_ 0\) are useful to deduce the existence of periodic solutions on prescribed energy surfaces. This is done in chapter 4, and, in particular, a detailed study of the Weinstein conjecture (a hypersurface \(S\) of contact type and satisfying \(H^ 1(S) = 0\) carries a closed characteristic) and related problems are presented.

Chapter 5 is devoted to the construction and study of a bi-invariant metric in the group of symplectic diffeomorphisms of the standard symplectic space, which are generated by time-dependent Hamiltonian vector fields having compactly supported Hamiltonian. This metric has nothing to do with Riemannian metrics on the infinite dimensional manifold of diffeomorphisms of \(\mathbb{R}^{2n}\).

The metric is closely related to the symplectic capacity and hence to periodic orbits. Moreover, another symplectic capacity, the so-called displacement energy, can be defined by means of \(d\).

Finally, a long appendix is included with technical topics presented separately.

All the chapters have a nice introduction with the historic development of the subject and with a perfect description of the state of the art. The main ideas are brightly exposed throughout the book.

This book, written by two experienced researchers, will certainly fill in a gap in the theory of symplectic topology. The authors have taken part in the development of such a theory by themselves or by their collaboration with other outstanding people in the area. Finall it is worth mentioning that the interest in this subject is growing at such a fast rate that another book has been published recently: “Introduction to Symplectic Topology”, by D. McDuff and D. Salamon, Oxford Mathematical Monographs, Clarendon Press, September (1994).

The origin of this theory can be traced back to Poincaré’s geometrical theorem (proved by Birkhoff): “Any area preserving map of the annulus on itself such that it moves its boundary circles on opposite directions, has at least two fixed points.”

But it was not until the advent of Arnol’d’s conjecture [V. I. Arnol’d, Russ. Math. Sur. 41, No. 6, 1-21 (1986); translation from Usp. Mat. Nauk 41, No. 6, 3-18 (1986; Zbl 0618.58021)] that symplectic topology has emerged as a subject of its own: If \(\phi\) is a nondegenerate exact symplectic diffeomorphism on a compact symplectic manifold, then \(\phi\) has at least as many fixed points as the Morse number of the manifold. Chapter 6 of the book under review deals with this conjecture within the spirit of Floer homology.

Reformulated dynamically, this conjecture gives a lower bound for the number of global forced oscillations of a time-dependent Hamiltonian vector field on a compact symplectic manifold in terms of the topology of this manifold, or in terms of the number of critical points for smooth functions on this manifold. The conjecture has been solved only for certain cases. For example, C. Conley and the second author [Invent. Math. 73, 33-49 (1983; Zbl 0516.58017)] proved it for the torus, and B. Fortune [Invent. Math. 81, 29-46 (1985; Zbl 0547.58015)] for the complex projective space. Later on, A. Floer [Duke Math. J. 53, 1-32 (1986; Zbl 0607.58016)] was able to prove the conjecture for symplectic manifolds with vanishing second homotopy group, and later on, for monotone symplectic manifolds [same author, J. Differ. Geom. 30, No. 1, 207-221 (1989; Zbl 0678.58012)]. Floer’s ideas are based on a fresh use of Morse theory as rephrased by E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)].

A combination of Floer’s construction and the special capacity \(c_ 0\) of chapter 3, allows the authors to present a symplectic homology theory. But this chapter is, in contrast to the previous chapters, not self- contained.

However, Arnol’d’s conjecture is not the central point of the book. The main theme is a class of symplectic invariants; the so-called symplectic capacities.

A symplectic capacity is a map \((M,\omega)\to c(M,\omega)\), where \((M,\omega)\) is a symplectic manifold possibly with boundary and of fixed dimension \(2n\), which assigns to each \((M,\omega)\) a nonnegative real number or \(\infty\) satisfying monotonicity, conformality and nontriviality. The main difference with volume symplectic invariants is that \(c(Z(1),\omega_ 0) < \infty\), where \(Z(1)\) is the open symplectic cylinder in the standard space. Chapter 2 is devoted to the axiomatic definition of symplectic capacities and to their application to embeddings. For example, Gromov’s well-known squeezing theorem is a consequence of the existence of symplectic capacities.

Chapter 3 outlines the existence proof of a distinguished capacity function, \(c_ 0\), that was introduced by the authors in Analysis et cetera, Res. Pap. in Honor of J. Moser’s 60th Birthd., 405-427 (1990; Zbl 0702.58021). This capacity measures the minimal \(C^ 0\)-oscillation of special Hamiltonian functions needed in order to conclude the existence of a distinguished periodic orbit having small period and solving the associated Hamiltonian system. The special features of \(c_ 0\) are useful to deduce the existence of periodic solutions on prescribed energy surfaces. This is done in chapter 4, and, in particular, a detailed study of the Weinstein conjecture (a hypersurface \(S\) of contact type and satisfying \(H^ 1(S) = 0\) carries a closed characteristic) and related problems are presented.

Chapter 5 is devoted to the construction and study of a bi-invariant metric in the group of symplectic diffeomorphisms of the standard symplectic space, which are generated by time-dependent Hamiltonian vector fields having compactly supported Hamiltonian. This metric has nothing to do with Riemannian metrics on the infinite dimensional manifold of diffeomorphisms of \(\mathbb{R}^{2n}\).

The metric is closely related to the symplectic capacity and hence to periodic orbits. Moreover, another symplectic capacity, the so-called displacement energy, can be defined by means of \(d\).

Finally, a long appendix is included with technical topics presented separately.

All the chapters have a nice introduction with the historic development of the subject and with a perfect description of the state of the art. The main ideas are brightly exposed throughout the book.

This book, written by two experienced researchers, will certainly fill in a gap in the theory of symplectic topology. The authors have taken part in the development of such a theory by themselves or by their collaboration with other outstanding people in the area. Finall it is worth mentioning that the interest in this subject is growing at such a fast rate that another book has been published recently: “Introduction to Symplectic Topology”, by D. McDuff and D. Salamon, Oxford Mathematical Monographs, Clarendon Press, September (1994).

Reviewer: J.Monterde (Burjasot)

##### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

53D40 | Symplectic aspects of Floer homology and cohomology |

54H20 | Topological dynamics (MSC2010) |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

70H30 | Other variational principles in mechanics |