Conjecture: Sharp equality for C^1 Axiom A systems on a compact domain

Prove that for C^1 Axiom A dynamical systems on a compact domain, the variational minimal-period bound max_{periodic orbits}(2π/T)^2 ≤ inf_{C>0, φ,V ∈ C^1(Ω)}{C : C|φ(x)|^2 − |ℒ_f φ(x)|^2 + ℒ_f V(x) ≥ 0 ∀ x ∈ Ω} holds as an equality.

Background

Building on the general sharp-equality question for the auxiliary-function method, the authors posit a specific setting—C^1 Axiom A systems on compact domains—where such equality may be expected. Axiom A systems are uniformly hyperbolic and structurally stable, features that often facilitate sharp ergodic and extremal properties.

A sharpness result exists for related SOS problems, but it does not directly apply to this context; thus, proving the conjectured equality would anchor the theoretical optimality of the method in a major class of dynamical systems.