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On the total disconnectedness of the quotient Aubry set


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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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