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On the total disconnectedness of the quotient Aubry set

Published online by Cambridge University Press:  01 February 2008

ALFONSO SORRENTINO
ALFONSO SORRENTINO*
Affiliation:
Department of Mathematics, Princeton University, Washington Road, Princeton, NJ 08544, USA (email: asorrent@math.princeton.edu)
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Abstract

In this paper we show that the quotient Aubry set, associated to a sufficiently smooth mechanical or symmetrical Lagrangian, is totally disconnected (i.e. every connected component consists of a single point). This result is optimal, in the sense of the regularity of the Lagrangian, as Mather’s counterexamples (J. N. Mather. Examples of Aubry sets. Ergod. Th. & Dynam. Sys.24(5) (2004), 1667–1723) show. Moreover, we discuss the relation between this problem and a Morse–Sard-type property for (the difference of) critical subsolutions of Hamilton–Jacobi equations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 1 , February 2008 , pp. 267 - 290
Copyright
Copyright © Cambridge University Press 2007

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