13 problems

Let $F$ be a twist map, let $c_1<c_2$, and let $\Lambda_{[c_1,c_2]}$ be the set of orbits corresponding to minimal configurations with $c\in[c_1,c_2]$. Let $\alpha$ denote the rele…

A Tonelli Lagrangian system is considered on $T\mathbb{T}^n$, with its minimizing Mather measure understood as a Borel probability measure on the tangent bundle. A Lagrangian is ca…

Let $H$ be a generic Hamiltonian with a unique Mather measure $\tilde\mu$. Mañé's conjecture. The unique Mather measure $\tilde\mu$ is supported on a closed curve. The source prese…

Let $M$ be a compact manifold, let $H$ be a Hamiltonian, and let $\mathcal A$ and $\mathcal M$ denote respectively the projected Aubry set and the projected Mather set associated w…

Let $M$ be a closed manifold and $L:TM\rightarrow \mathds{R}$ a Tonelli Lagrangian. For some $r\geq 2$, consider perturbations of $L$ by $C^r$-generic potentials. Maf1e's conjectur…

A Tonelli Lagrangian on a closed manifold $M$ determines a Mather set in the tangent bundle $TM$. Mañé's conjecture. For a generic Tonelli Lagrangian on $M$, the Mather set is redu…

Extension conjecture. The same approach should extend to other settings in which Aubry–Mather theory has been applied, including monotone problems on lattices and elliptic partial…

Large-mean-forcing integrability conjecture. For fixed $V$ and $\sigma(F)$, there is an $F_{c}$ such that, for all $\overline{F}\geq F_{c}$, the map $f_{V,F}$ is integrable.

Extension conjecture. All the results obtained for the discrete-space system … .

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 $C^\infty(M)$.

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…