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Vertices of Mather’s beta function, II

Published online by Cambridge University Press:  01 August 2009

DANIEL MASSART
DANIEL MASSART*
Affiliation:
Département de Mathématiques, Université Montpellier 2, 34095 Montpellier Cedex 5, France (email: [email protected])
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Abstract

If the β-function of a time-periodic Lagrangian on a manifold M has a vertex at a k-irrational homology class h, then 2k<dim M, so if dim M=2, then h is rational.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1289 - 1307
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Auer, F. and Bangert, V.. Differentiability of the stable norm in codimension one. Amer. J. Math. 128(1) (2006), 215–238.CrossRefGoogle Scholar
[2]Bangert, V.. Geodesic rays, Busemann functions and monotone twist maps. Calc. Var. Partial Differential Equations 2(1) (1994), 49–63.CrossRefGoogle Scholar
[3]Bernard, P.. Connecting orbits of time dependent Lagrangian systems. Ann. Inst. Fourier (Grenoble) 52(5) (2002), 1533–1568.CrossRefGoogle Scholar
[4]Bernard, P.. The dynamics of pseudographs in convex Hamiltonian systems. J. Amer. Math. Soc. 21(3) (2008), 615–669.CrossRefGoogle Scholar
[5]Burago, D., Ivanov, S. and Kleiner, B.. On the structure of the stable norm of periodic metrics. Math. Res. Lett. 4(6) (1997), 791–808.CrossRefGoogle Scholar
[6]Carneiro, M. J. D.. On minimizing measures of the action of autonomous Lagrangians. Nonlinearity 89 (1995), 1077–1085.CrossRefGoogle Scholar
[7]Fathi, A.. Weak KAM Theorem in Lagrangian Dynamics. Cambridge University Press, Cambridge, to appear.Google Scholar
[8]Fathi, A., Figalli, A. and Rifford, L.. On the hausdorff dimension of the mather quotient. Commun. Pure Appl. Math. to appear.Google Scholar
[9]Ferry, S.. When ϵ-boundaries are manifolds. Fund. Math. (3) (1975/76), 199–210.Google Scholar
[10]Hocking, J. and Young, G.. Topology, 3rd edn. Dover, New York, 1988.Google Scholar
[11]Mañé, R.. On the minimizing measures of Lagrangian dynamical systems. Nonlinearity 5(3) (1992), 623–638.Google Scholar
[12]Mañé, R.. Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9(2) (1996), 273–310.CrossRefGoogle Scholar
[13]Massart, D.. Stable norms of surfaces: local structure of the unit ball at rational directions. Geom. Funct. Anal. 7(6) (1997), 996–1010.CrossRefGoogle Scholar
[14]Massart, D.. On Aubry sets and Mather’s action functional. Israël J. Math. 134 (2003), 157–171.Google Scholar
[15]Massart, D.. Subsolutions of time-periodic Hamilton–Jacobi equations. Ergod. Th. & Dynam. Sys. 27(4) (2007), 1253–1265.Google Scholar
[16]Massart, D.. Aubry sets vs Mather sets in two degrees of freedom. Preprint arXiv:0803.2647v1 [math.DS].Google Scholar
[17]Mather, J. N.. Differentiability of the minimal average action as a function of the rotation number. Bol. Soc. Brasil. Mat. (N.S.) 21(1) (1990), 59–70.Google Scholar
[18]Mather, J. N.. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991), 169–207.CrossRefGoogle Scholar
[19]Mather, J. N.. A property of compact, connected, laminated subsets of manifolds. Ergod. Th. & Dynam. Sys. 22(5) (2002), 1507–1520.CrossRefGoogle Scholar
[20]Osuna, O.. Vertices of Mather’s beta function. Ergod. Th. & Dynam. Sys. 25(3) (2005), 949–955.CrossRefGoogle Scholar