Renormalization for Bruin–Troubetzkoy ITMs

The paper proves there exists a renormalization-invariant probability µ supported on the full non-escaping set G, and that for µ-almost every parameter the Bruin–Troubetzkoy 3-ITM is uniquely ergodic (Theorem 3), by encoding a new induction R as a simplicial (MCF) system, verifying quickly-escaping/expanding hypotheses, applying Fougeron’s theorem to obtain a natural measure, and concluding unique ergodicity via Veech’s standard argument . The induction is explicitly three-cased with matrices A and CA in the (1,2,3) permutation and, symmetrically, B and CB in the (2,1,3) permutation, yielding a two-vertex Rauzy-like graph; an acceleration recovering the classical Bruin–Troubetzkoy Gauss map is also exhibited . The candidate solution outlines essentially the same approach: encode the renormalization as a Markovian MCF, perform a special acceleration to obtain positive matrices (the paper’s Lemma 11), invoke Fougeron’s simplicial-systems framework to construct an invariant measure supported on G, and obtain unique ergodicity a.e. by the standard contraction/tower argument. Differences are mostly expository: the paper’s R uses four matrices (A,B,CA,CB) and a two-vertex graph, whereas the candidate restricts to products of A and CA (implicitly working on the accelerated, single-vertex section that recovers the Gauss map); and the candidate implicitly assumes integrability of return times in the suspension step rather than citing Fougeron’s theorem directly. These are minor and do not affect correctness. Overall, the model’s proof sketch aligns with the paper’s method and conclusions .