Finite codimension stability of invariant surfaces

The paper proves finite-codimension stability of typical invariant higher-genus surfaces for flat geodesic flows on translation surfaces via a para-differential, fixed-point approach (Theorem 1.1, para-linearization, and a contraction mapping with counterterms) and establishes linear growth of the codimension in s and in the topological complexity; the candidate solution proves the same statement by a Nash–Moser scheme anchored in the cohomological equation for translation flows and a finite family of obstruction-killing conditions. The two arguments align on hypotheses (almost every direction; equality to H0 near Σ), the invariant-graph conclusion, and the linear codimension growth, but use substantially different techniques. See Theorem 1.1 and the para-linearization/linear-solve steps in the paper for the para-differential route , and the cohomological equation framework underpinning both approaches .