Mañé, Ricardo On the minimizing measures of Lagrangian dynamical systems. (English) Zbl 0799.58030 Nonlinearity 5, No. 3, 623-638 (1992). Summary: We consider dynamical systems generated by time-dependent periodic Lagrangians on a closed manifold \(M\). An invariant probability \(\mu\) of such a system has a homology \(\rho_ n(\mu)\) element of \(H_ 1(M,R)\) describing (roughly speaking) the average homological position of \(\mu\) a.e. orbit, and an action \(A(\mu)\) defined as the integral with respect to \(\mu\) of the Lagrangian. The minimizing measures are defined as the invariant probabilities of the system that minimize the action among those that have a given homology \(\lambda\). Their action \(\beta(\gamma)\) defines a convex function \(\beta : H_ 1(M,R) \to R\). These concepts were introduced by Mather where he proved several theorems about them. In this paper we further develop Mather’s theory, giving a characterization of minimizing measures and proving properties about minimizing measures whose homologies are strictly extremal points of the function \(\beta\). Cited in 1 ReviewCited in 40 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 37A99 Ergodic theory Keywords:minimizing measures; Lagrangian dynamical systems PDFBibTeX XMLCite \textit{R. Mañé}, Nonlinearity 5, No. 3, 623--638 (1992; Zbl 0799.58030) Full Text: DOI