12 problems

Let a Lagrangian be chosen from the generic class considered in the ergodic optimization setting, and let minimizing measures be invariant probability measures minimizing the space…

Man'e's conjecture. For a generic Lagrangian $L$ on a closed manifold $M$, there exist a dense open set $U_0$ of $H^1(M,\mathbb{R})$ such that for every $\omega\in U_0$,…

Let a minimal ergodic measure for an autonomous Lagrangian system be given in the homotopical version. A finite-fold covering space is a covering space with finitely many sheets, a…

Consider an integral curve $\gamma$ determined by intersections of smooth branches indexed by $i\in\mathfrak I$, and let $\mathfrak J_3,\mathfrak J_4,\ldots$ be the successively de…

Made's generic Aubry-orbit conjecture. If $L$ is an autonomous Tonelli Lagrangian on $M$, there is a residual subset $\mathcal O_4(L)\subset C^\infty(M)$ such that, for every…

Made's Aubry-set conjecture. The set $\mathcal O(L)$ is residual in $C^\infty(M)$.

Made's weak conjecture. For a generic Lagrangian $L$, there is an open dense subset $U$ of $H^1(M,\mathbb R)$ such that, for every $c\in U$, there is a unique $(L,c)$-minimizing me…

Made's strong conjecture. For a generic Lagrangian, there is a unique minimizing measure, and this measure is supported on a periodic orbit or a fixed point of the Euler–Lagrange f…

Let $M$ be a manifold and let $L$ be a Tonelli Lagrangian on $TM$. For a cohomology class $c\in H^1(M,\mathbb{R})$, let the Aubry set of $(L+f,c)$ denote the Aubry set associated w…

Mañé's Aubry-set conjecture. The set $\mathcal{O}_3(L)$ is residual in $C^{\infty}(M)$. The source states that this conjecture is equivalent to the strong conjecture on generic min…

Let $M$ be a manifold and let $L$ be a Tonelli Lagrangian on $TM$. For a cohomology class $c\in H^1(M,\mathbb{R})$, write $\mathcal{M}(L,c)$ for the Mather set associated with $L$…

Mañé's strong conjecture. The set $\mathcal{O}_1(L)$ is residual in $C^{\infty}(M)$. Equivalently, for a generic potential perturbation, there is a unique minimizing measure and it…