Mañé’s conjecture on the generic structure of the Aubry set

Prove Mañé’s conjecture that, generically, the Aubry set of a Tonelli Hamiltonian is either a hyperbolic equilibrium or a hyperbolic periodic orbit, thereby establishing the generic applicability of the linear O(λ) vanishing-discount convergence rate.

Background

The paper proves an O(λ) convergence rate for the vanishing-discount problem when the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit. It then relates this result to Mañé’s conjecture, which predicts that this single-component hyperbolic structure is generic. Establishing the conjecture would therefore imply an O(λ) vanishing-discount convergence rate for generic Tonelli Hamiltonians.

References

Corollary \ref{cor:single-intro} is also closely related to Mañé's conjecture, which predicts that, generically, the Aubry set is either a hyperbolic equilibrium or a hyperbolic periodic orbit; see and the references therein.

— Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets  (2609.02779 - Ni, 2 Sep 2026) in Section 1, Introduction, immediately following Corollary \ref{cor:single-intro}