Finiteness and coexistence of maximizing rotation vectors in higher dimensions

Determine whether the set of globally maximizing rotation vectors is finite for every inverse temperature in primitive finite-state correspondences with at least two observables satisfying the real-analytic Legendre assumptions, and characterize the parameter values at which coexistence occurs.

Background

The paper proves finiteness of nonlinear equilibrium values and pairs in one dimension under real-analytic Legendre regularity and uniqueness of linear equilibrium pairs. It explicitly notes that the isolated-zero argument does not extend to several observables, because real-analytic critical sets in dimensions greater than one need not be discrete.

The unresolved problem concerns the quadratic energy F_β(z)=β||z||²/2 for primitive finite-state correspondences with d≥2 observables. The three-state Potts example demonstrates that coexistence can occur, but does not establish a general finiteness theorem or classify the temperatures at which multiple globally maximizing rotation vectors coexist.