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A construction of a large family of integrable symplectic birational maps Suris, Yuri


We give a construction of completely integrable (2m)-dimesnional Hamiltonian systems with m cubic integrals in involution. Applying to the corresponding quadratic Hamiltonian vector fields the so called Kahan-Hirota-Kimura discretization scheme, we arrive at birational (2m)-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on $R^{2m}$, and possess m independent integrals of motion, which are perturbations of the original integrals. Thus, these maps are completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original m-tuples of commuting vector fields, the m-tuples of maps commute and share the invariant symplectic structure and m integrals of motion.

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