Geometric Markov Partitions for Pseudo-Anosov Homeomorphisms with Prescribed Combinatorics

The paper defines adapted Markov partitions (only periodic boundary points are singularities, and each such point is a corner) and proves their existence via compatible adapted graphs; Proposition 6 yields an adapted Markov partition directly, and Construction 1 + Corollary 2 ensure every generalized pseudo‑Anosov admits such partitions, while Theorem 8 provides a corner refinement algorithm that preserves boundary periodic points . The candidate solution reaches the same end by a classical route: build a foliation‑aligned Markov partition (Bowen‑style refinement) and then slide sides along leaves to remove all non‑singular periodic boundary points, leaving only singularities at corners. Results coincide; proofs are substantively different.