Index theory for symplectic paths with applications. (English) Zbl 1012.37012

Progress in Mathematics (Boston, Mass.). 207. Basel: Birkhäuser. xxiv, 380 p. (2002).
In order to briefly describe the subject of this monograph recall that a real \(2n\times 2n\)-matrix \(M\) is called symplectic if \(M^TJM=J\), where \(J\) is equal to \(\left[\begin{smallmatrix} 0&-I\\ I&0 \end{smallmatrix}\right]\). The space of symplectic matrices is denoted by \(\text{Sp}(2n)\) and its codimension-one subspace consisting of matrices \(M\) such that \(\omega\) is its eignenvalue is denoted by \(\text{Sp}(2n)_\omega^0\). For \(\tau>0\), the space of symplectic matrix paths is defined as \[ P_\tau(2n):=\{\gamma\in C([0,\tau],\text{Sp}(2n))\colon \gamma(0)=I\}. \] A symplectic matrix path arises naturally as the fundamental matrix of a linear Hamiltonian system \(\dot x=JB(t)x\) for some matrix-valued function \(B\), and the theory of Hamiltonian systems motivated introduction of invariants related to symplectic paths. Let \(\gamma\in P_\tau(2n)\). For every element \(\omega\) of the unit circle in \(\mathbb{C}\), the index with respect to \(\omega\) of \(\gamma\) is the pair \[ (i_\omega(\gamma),\nu_\omega(\gamma))\in {\mathbb{Z}}\times \{0,\ldots,2n\}, \] where \(\nu_\omega(\gamma)\), called the nullity, is equal to the dimension (over \(\mathbb{C}\)) of the kernel of \(\gamma(\tau)-\omega I\), and \(i_\omega(\gamma)\), called the rotation, can be described in the nondegenerate case \(\nu_\omega(\gamma)=0\) as the intersection number of the joint path \(\gamma\ast\xi\) and \(\text{Sp}(2n)_\omega^0\), where \(\xi\) connects the diagonal symplectic matrix having a sequence of \(2\)-s and \(1/2\)-s as the diagonal entries, with \(I\). The index was introduced in the paper of C. Conley and E. Zehnder [Commun. Pure Appl. Math. 37, 207-253 (1984; Zbl 0559.58019)] in the case of nondegenerate paths, \(\omega=1\), and \(n\geq 2\), and it was further extended in papers of Y. Long, E. Zehnder, C. Viterbo, J. Robbin, and D. Salamon. The presented in the monograph final version together with axiomatic characterization comes from the paper by Y. Long [Topol. Methods Nonlinear Anal. 10, No. 1, 47-78 (1997; Zbl 0977.53075)]. The monograph provides construction, properties and applications of the index and related invariants. In Sections 1 and 2, an algebraic and topological aspects of the group \(\text{Sp}(2n)\) and its subgroups are presented. A particular strength is put on the geometric description in the case \(n=1\). An information on basic definitions and results on variational methods is provided in Sections 3 and 4. The index is defined in Section 5 and its properties and relations to the Morse theory are given in Sections 6 and 7. In the following Sections 8-12 problems related to iterations of symplectic paths are considered. In particular, they include generalizations of the Bott’s formulas on iterations of closed geodesics. The last three sections are devoted to applications of the index to some very important problems of nonlinear analysis. In Section 13, results related to the Rabinowitz conjecture on minimal periods of periodic solutions of Hamiltonian systems are given, Section 14 provides results on the Conley conjecture on the number of periodic solutions of periodic Hamiltionan systems on the \(2n\)-dimensional torus, and Section 15 geometrically distinct closed characteristics on compact strictly convex hypersurfaces of \(\mathbb{R}^{2n}\).


37B30 Index theory for dynamical systems, Morse-Conley indices
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
70H05 Hamilton’s equations
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
70H03 Lagrange’s equations
53D12 Lagrangian submanifolds; Maslov index
34C25 Periodic solutions to ordinary differential equations