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Hamiltonian diffeomorphisms and Lagrangian distributions. (English) Zbl 0761.58010

The authors study invariant Lagrangian submanifolds of a symplectic manifold in the presence of a Lagrangian distribution (a field of Lagrangian subspaces). One of the main subjects of the paper is the investigation of the interaction between the special class of optical Hamiltonian flows and the Maslov class, both of which are well-defined in the case under consideration. The observed non-trivial phenomena are closely related to classical Birkhoff’s theory for area preserving diffeomorphisms of the cylinder and may be considered as a generalization of the latter. Part 1 of the paper (Sections 1-7) is devoted to geometrical and dynamical properties of optical Hamiltonian diffeomorphisms on symplectic manifold endowed with a Lagrangian distribution. In Part 2 (Sections 1-3) certain groups of Hamiltonian diffeomorphisms and their actions are studied.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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