Solenoids of Split Sequences

The paper’s Theorem 6.3 states that for a strongly proper expanding split sequence, the cone TM(X) of transverse measures is the inverse limit of levelwise nonnegative cones with bonding maps given by the transition matrices of the folds; the detailed proof in Chapter 6 proceeds via “edge weights,” “weight assignments,” and turn-transversal decompositions, and uses expansion to ensure non-atomicity and appropriate local product structures . The candidate model solution proves the same identification but via a slightly different construction based on flow boxes, clopen elementary transversals in fibers, and one-step holonomy, deriving the same compatibility relation w_{j-1}=M(f_j)w_j and the inverse construction. The only substantive gaps in the model sketch are routine technicalities: justifying that every compact transversal breaks into finitely many elementary pieces and clarifying the precise role of the expansion/non-atomicity hypothesis (the paper addresses these via Proposition 5.15 and its turn-based machinery) . Overall, both arguments are consistent and correct; the approaches differ in presentation and technical scaffolding, not in substance.