Skip to main content Accessibility help
×
×
Home

On the total disconnectedness of the quotient Aubry set

  • ALFONSO SORRENTINO (a1)
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.

Copyright
References
Hide All
[1]Abraham, R. and Robbin, J.. Transversal Mappings and Flows. W. A. Benjamin, New York, 1967, With an appendix by Al Kelley.
[2]Bangert, V.. Mather sets for twist maps and geodesics on tori. Dynamics Reported, Vol. 1 (Dynam. Report. Ser. Dynam. Systems Appl., 1). Wiley, Chichester, 1988, pp. 156.
[3]Bates, S. M.. Toward a precise smoothness hypothesis in Sard’s theorem. Proc. Amer. Math. Soc. 117(1) (1993), 279283.
[4]Bernard, P.. Existence of C 1,1 critical sub-solutions of the Hamilton–Jacobi equation on compact manifolds. Ann. Sci. École Norm. Sup. (4), to appear.
[5]Burago, D., Ivanov, S. and Kleiner, B.. On the structure of the stable norm of periodic metrics. Math. Res. Lett. 4(6) (1997), 791808.
[6]Contreras, G., Delgado, J. and Iturriaga, R.. Lagrangian flows: the dynamics of globally minimizing orbits. II. Bol. Soc. Brasil. Mat. (N.S.) 28(2) (1997), 155196.
[7]Contreras, G. and Iturriaga, R.. Global minimizers of autonomous Lagrangians. 22 Colóquio Brasileiro de Matemática (22nd Brazilian Mathematics Colloquium). Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999.
[8]Fathi, A.. Sard, Whitney, Assouad and Mather. Talk at Recent and Future developments in Hamiltonian Systems (Institut Henri Poincaré, Paris, France, May 2005).
[9]Fathi, A.. Weak KAM Theorem and Lagrangian Dynamics. Cambridge University Press, Cambridge, to appear.
[10]Fathi, A. and Siconolfi, A.. Existence of C1 critical subsolutions of the Hamilton–Jacobi equation. Invent. Math. 155(2) (2004), 363388.
[11]Forni, G. and Mather, J. N.. Action minimizing orbits in Hamiltonian systems. Transition to Chaos in Classical and Quantum Mechanics (Montecatini Terme, 1991) (Lecture Notes in Mathematics, 1589). Springer, Berlin, 1994, pp. 92186.
[12]Hajlasz, P.. Whitney’s example by way of Assouad’s embedding. Proc. Amer. Math. Soc. 131(11) (2003), 34633467 (electronic).
[13]Mañé, R.. Lagrangian flows: the dynamics of globally minimizing orbits. Bol. Soc. Brasil. Mat. (N.S.) 28(2) (1997), 141153.
[14]Mather, J. N.. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207(2) (1991), 169207.
[15]Mather, J. N.. Variational construction of connecting orbits. Ann. Inst. Fourier (Grenoble) 43(5) (1993), 13491386.
[16]Mather, J. N.. Total disconnectedness of the quotient Aubry set in low dimensions. Comm. Pure Appl. Math. 56(8) (2003), 11781183. Dedicated to the memory of Jürgen K. Moser.
[17]Mather, J. N.. Examples of Aubry sets. Ergod. Th. & Dynam. Sys. 24(5) (2004), 16671723.
[18]Whitney, H.. Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36(1) (1934), 6389.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed