Markov partitions for non-transitive expansive flows

The paper proves (in two explicit steps) the existence of a Markov partition for non-transitive pseudo-Anosov flows with stable/unstable boundaries contained in the unions of weak stable/unstable manifolds of a finite set Γ, using standard polygons, a disjointization-by-flow trick, and a boundary-invariance criterion (Lemma 8) to certify the Markov property . By contrast, the model’s outline omits the crucial boundary-avoidance argument (it neither constructs the positively/negatively invariant Ks_γ, Ku_γ nor checks f(∪∂sRi) and f^{-1}(∪∂uRi) avoid interiors), and it assumes uniform continuity of hitting times along plaques without supplying the technical lemmas the paper proves or cites. Hence the paper’s proof is correct, but the model’s is incomplete as a proof of the stated theorem .