DYNAMICS ON MARKOV SURFACES: CLASSIFICATION OF STATIONARY MEASURES

The paper’s Main Theorem classifies all µ-stationary ergodic measures for the Vieta-involution action on the Markov/relative character surfaces S(A,B,C,D): every such measure has compact support, is ΓS-invariant, and is either the uniform measure on a finite ΓS-orbit or the normalized symplectic area on the unique compact real component S(R)c when A,B,C,D are real and in [-2,2] (with S(R)c a 2-sphere or a point). This is stated and proved in Section 1.4 and Section 6 (Main Theorem; Proposition 6.1; Corollary 6.3), using the invariant area form (Area) and a combination of dynamics at infinity and random-dynamical measure rigidity (Ledrappier; Brown–Rodriguez–Hertz) . The candidate solution’s final classification agrees with the paper, but its proof contains critical gaps and misstatements: (i) it asserts an “iff” condition for the existence of S(R)c in terms of A,B,C,D ∈ [-2,2]^4 that the paper only proves as a necessary condition (the topology is controlled via boundary traces a,b,c,d, and the compact component is unique when it exists) ; (ii) it claims the mapping-class image has index ≤6, whereas the paper quotes index ≤24 for the relevant action generated by the Vieta involutions ; (iii) it invokes a global Margulis/drift function giving P M ≥ M + c outside a compact set without proof, while the paper derives compactness of support by a precise boundary-dynamics/“two vertices at infinity” argument (Proposition 6.1) ; and (iv) it sketches “stationary ⇒ invariant” without the measure-rigidity mechanisms (Ledrappier’s invariance principle and Brown–Rodriguez–Hertz) that the paper uses to close the proof on S(R)c (Section 6.4–6.5) . Hence, while the model’s conclusion matches the paper, the provided proof is not correct as written.