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Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom(AMS-208)$
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Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom: (AMS-208)

Vadim Kaloshin and Ke Zhang

Print publication date: 2020

Print ISBN-13: 9780691202525

Published to Princeton Scholarship Online: May 2021

DOI: 10.23943/princeton/9780691202525.001.0001

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Aubry-Mather type at the single resonance

Aubry-Mather type at the single resonance

Chapter:
(p.92) (p.93) Chapter Nine Aubry-Mather type at the single resonance
Source:
Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom
Author(s):

Kaloshin Vadim

Zhang Ke

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691202525.003.0009

This chapter proves Aubry-Mather type in the single-resonance regime. It proves that for a single-resonance normal form system which satisfies the non-degeneracy conditions, every c in the resonance curve is of either Aubry-Mather or bifurcation Aubry-Mather type. The main results are Theorems 9.3 and 9.5, which restate Propositions 3.9 and 3.10. The chapter then proves that the conditions hold on an open and dense set of Hamiltonians. It discusses bifurcations in the double maxima case, as well as hyperbolic coordinates. The chapter also examines normally hyperbolic invariant cylinder, the localization of the Aubry and Mañé sets, and the genericity of the single-resonance conditions.

Keywords:   Aubry-Mather type, single-resonance regime, single-resonance normal form, bifurcation Aubry-Mather type, Hamiltonians, hyperbolic coordinates, normally hyperbolic invariant cylinder, Aubry sets, Mañé sets

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