Abstract
New 2-dimensional Birkhoff surfaces of sections are defined for the desymmetrized P SL(2, Z) group. The new object is demonstrated to be apt to study the geodesics flow solution of the Hamiltonian problem. The new definitions of the return maps of the new 2-dimensional Birkhoff surface of section are provided with; The demonstration relays on the self-adjointed-ness of the operators on which the conjugacy subclasses needed in the application to reduced surds act
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