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Repulsion from Resonances
Dmitry Dolgopyat, University of Maryland, College Park, MD
A publication of the Société Mathématique de France.
Mémoires de la Société Mathématique de France
2012; 119 pp; softcover
Number: 128
ISBN-10: 2-85629-344-1
ISBN-13: 978-2-85629-344-7
List Price: US$48
Member Price: US$38.40
Order Code: SMFMEM/128
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The author considers slow-fast systems with periodic fast motion and integrable slow motion in the presence of both weak and strong resonances. Assuming that the initial phases are random and that appropriate non-degeneracy assumptions are satisfied, he proves that the effective evolution of the adiabatic invariants is given by a Markov process. This Markov process consists of the motion along the trajectories of a vector field with occasional jumps. The generator of the limiting process is computed from the dynamics of the system near strong resonances.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians interested in averaging, slow-fast systems, Markov processes, invariant cones, and resonances.

Table of Contents

  • Introduction
  • The proof
  • A. Asymptotics of the Poincaré map
  • B. Derivatives of the Poincaré map. Outline of the proof
  • C. Derivatives of the inner map
  • D. Derivatives of the outer map
  • E. Dynamics near the separatrix
  • F. Captured points
  • G. Examples
  • H. Distortion bound
  • Bibliography
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