Critical Invariant Polygons and the Farey--Ito Boundary of Stochastic Spectra

Abstract

Let $T$ be an orientation-preserving real-linear contraction of an oriented plane with nonreal eigenvalues, and let its polygonal complexity be the least number of vertices of a nondegenerate convex polygon $P$ satisfying $TP\subseteq P$. We first develop an intrinsic contact theory for radially critical maps. Hereditary saturation, one-sided contact selection, exact contact mutations, a finite cyclic reduction, and the unimodular lattice sail lead to a contact-return normal form and a projective no-skipping principle. In an adapted complex coordinate, the resulting monodromy yields a heterogeneous Ito product and an exact lifted phase identity carried by Farey neighbors. We then apply this structure to determine, for every $n$, the union $Θ_n$ of the spectra of all real row-stochastic $n\times n$ matrices. A strictly convex log-sine potential equalizes the monodromy factors and gives the sharp radial equation. Sparse stochastic matrices attain every equality point, while scalar Farey refinement proves nesting from order $n-1$ to order $n$. Together with radial filling and the unit-circle analysis, this establishes the full Farey--Ito description of the Karpelevič region from critical invariant polygons.