Abstract:
In 1984 V. F. Lazutkin [Laz84, LST89] obtained an asymptotic formula for the separatrix splitting angle for the standard map. The difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter. Lazutkin's proof was based on two conjectures. Probably, the original form of those conjectures was incorrect, but Lazutkin's method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices. The consequent works [Laz91, Laz92, GLS94] and [Gel96] prepared the base to fill all the gaps of the original proof. The present paper contains a complete and self-contained proof of a refined version of the original formula (formula (1.7) of the present paper). In this form the formula was obtained in [GLS94]. The proof is inspired by the ideas of Lazutkin's original paper [Laz84].
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Received: 16 February 1998 / Accepted: 20 August 1998
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Gelfreich, V. A Proof of the Exponentially Small Transversality of the Separatrices for the Standard Map. Comm Math Phys 201, 155–216 (1999). https://doi.org/10.1007/s002200050553
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DOI: https://doi.org/10.1007/s002200050553