Vertices of Mather’s beta function, II

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.

A dynamical condition for differentiability of Mather's average action

“…a graph which not contain ω-limit of minimizing curves out of it 1 . More precisely, we prove the following theorem: We prove the converse of Theorem 2 in the case M equals torus T 2 (see Theorem 15). In this case, the set V = c∈U 0 ∂α (c) , obtained in the above statement, is open in H 1 (T 2 ; R).…”

“…Understanding whether or not this function is differentiable and what are the implications of its regularity to the dynamics of the system is an interesting problem. This type of problem was developed by D. Massart in several works as, for example, [14] and [15].…”

Mañé's conjectures in codimension 1

“…We define rat(ρ, 1) as the subspace of R n generated by Z n × Z ∩ (ρ, 1) ⊥ ; we also define M (ρ) as the projection on R n of rat(ρ, 1). Recall from [Mt09] (Proposition 18) that the irrationality I Z (ρ) of ρ is the dimension of A(ρ). The next lemma implies that I Z (ρ) = n − dim M (ρ) = n − dim rat(ρ, 1).…”

Systèmes lagrangiens et fonction $β$ de Mather

“…Soit I R (h) la dimension de l'adhérence dans T b de Rh au lieu de Zh. Sur la relation entre les deux notions d'irrationalité, voir l'appendice A.1 de [Mt09]. Notons que la fonction I R (h) est zero-homogène, i.e.…”

“…Avertissement : ce texte vise à présenter les résultats de [Mt03], [Mt07], [BaM08], [Mt09], [Mta], [MS], [Mt10], [BeM], et [Mtb]. Il ne constitue pas une introduction au sujet, ce pour quoi on consultera avec davantage de profit les références [Ba88], [MrF91], [F], et [Mr09].…”