Minimal measures for Euler-Lagrange flows on finite covering spaces

Fang Wang, Zhihong Xia

Research output: Contribution to journalArticle

Abstract

In this paper we study the minimal measures for positive definite Lagrangian systems on compact manifolds. We are particularly interested in manifolds with more complicated fundamental groups. Mather's theory classifies the minimal or action-minimizing measures according to the first (co-)homology group of a given manifold. We extend Mather's notion of minimal measures to a larger class for compact manifolds with non-commutative fundamental groups, and use finite coverings to study the structure of these extended minimal measures. We also define action-minimizers and minimal measures in the homotopical sense. Our program is to study the structure of homotopical minimal measures by considering Mather's minimal measures on finite covering spaces. Our goal is to show that, in general, manifolds with a non-commutative fundamental group have a richer set of minimal measures, hence a richer dynamical structure. As an example, we study the geodesic flow on surfaces of higher genus. Indeed, by going to the finite covering spaces, the set of minimal measures is much larger and more interesting.

Original languageEnglish (US)
Pages (from-to)3625-3646
Number of pages22
JournalNonlinearity
Volume29
Issue number12
DOIs
StatePublished - Oct 14 2016

Fingerprint

Covering Space
Lagrange
Euler
coverings
Fundamental Group
Compact Manifold
Lagrangian Systems
Geodesic Flow
Homology Groups
homology
Minimizer
Positive definite
Genus
Covering
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Keywords

  • Hamiltonian mechanics
  • Lagrangian systems
  • Mather theory
  • geodesic flow

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

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Minimal measures for Euler-Lagrange flows on finite covering spaces. / Wang, Fang; Xia, Zhihong.

In: Nonlinearity, Vol. 29, No. 12, 14.10.2016, p. 3625-3646.

Research output: Contribution to journalArticle

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