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Denjoy subsystems and horseshoes

Published online by Cambridge University Press:  10 May 2024

By
Marie-Claude Arnaud
Edited by
Albert Fathi ,
Philip J. Morrison ,
Tere M-Seara  and
Sergei Tabachnikov
Albert Fathi
Affiliation:
Georgia Institute of Technology
Philip J. Morrison
Affiliation:
University of Texas, Austin
Tere M-Seara
Affiliation:
Universitat Politècnica de Catalunya, Barcelona
Sergei Tabachnikov
Affiliation:
Pennsylvania State University
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Summary

We introduce a notion of weak Denjoy subsystem (WDS) that generalizes the Aubry–Mather–Cantor sets to diffeomorphisms of manifolds. We explain how a rotation number can be associated to such a WDS. Then we build in any horseshoe a continuous one parameter family of such WDS that is indexed by its rotation number. Looking at the inverse problem in the setting of Aubry– Mather theory, we also prove that for a generic conservative twist map of the annulus, the majority of the Aubry–Mather sets are contained in some horseshoe that is associated to an Aubry–Mather set with a rational rotation number.

Keywords

smooth mappings and diffeomorphismssymbolic dynamicshomoclinic and heteroclinic orbitshyperbolic orbits and setstwist mapsrotation numbers and vectors
Type
Chapter
Information
Hamiltonian Systems
Dynamics, Analysis, Applications
 , pp. 1 - 28
Publisher: Cambridge University Press
Print publication year: 2024

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