Dynamical Systems Around the Rauzy Gasket and Their Ergodic Properties

The paper’s Theorem 1.1 proves that the four constructions—(a) the mapping torus (Σ_AR_λ,F_AR_λ), (b) the RP^2 branched-cover model (Σ_λ,F_λ), (c) the PL model in T^3 (Σ_PL,F^PL_λ), and (d) the double suspension Σ(S_λ,θ_λ)—all yield equivalent measured foliations for λ∈Δ\∂Δ, by exhibiting closed transversals whose first-return map is TAR_λ and invoking direct identifications (Propositions 5.1, 6.1, 8.1) . The candidate solution follows the same blueprint: a reconstruction-from-return-data lemma and explicit transversals realizing the same TAR_λ (with the six lengths λ_i/2 and the rotation by 1/2) , then concluding equivalence. Minor differences: the paper states exact isomorphisms for (a)↔(b) and (a)↔(c) (no rescaling), and an explicit factor 2 for (a)↔(d) , whereas the model allows unspecified global scalings and suggests a covering-degree scaling in (b), which is unnecessary. Overall, both arguments are materially the same and correct; the model’s lemma is the standard cut-and-glue/zippered-rectangle reconstruction implicitly used in the paper.