Viterbo, Claude Symplectic topology as the geometry of generating functions. (English) Zbl 0735.58019 Math. Ann. 292, No. 4, 685-710 (1992). We introduce a very simple approach to symplectic topology, using the notion of generating functions quadratic at infinity of Lagrange submanifolds in a cotangent bundle. We first prove that such a generating function is essentially unique. Then, certain critical values of the generating function yields invariants of the Lagrange submanifold. We prove a certain number of properties of these invariants under deformation and ”composition”. If our Lagrange submanifold is the graph of a compact supported symplectic map \(\psi\) of \(\mathbb{R}^{2n}\) we get two numbers \(c_ +(\psi)\) and \(c_ -(\psi)\). We prove many properties of these invariants, \(C^ 0\) continuity, continuity with respect to a generating Hamiltonian etc.…. These invariants are used to get estimates on the number of periodic orbits and to define symplectic invariants for sets. The symplectic invariants thus defined satisfy the axioms of capacities. They also behave very nicely with respect to symplectic reduction, thus allowing us to give simple proofs of the ”camel problem”, and study properties of ”simple hypersurfaces”. Reviewer: C.Viterbo Cited in 14 ReviewsCited in 144 Documents MSC: 53D35 Global theory of symplectic and contact manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension Keywords:symplectic topology; generating functions; invariants; Lagrange submanifold; periodic orbits; symplectic invariants for sets × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Arnold, V.I.: First steps of symplectic topology. Russ. Math. Surv.6, 3-18 (1986) · Zbl 0618.58021 [2] Cerf, J.: La stratification naturelle des espaces de fonctions diff?rentiables r?elles et le th?or?me de la pseudo-isotopie. Publ. Math., Inst. Hautes ?tud. 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