Joint invariant sets for non-commutative expanding Markov maps of the circle

The paper proves a global generic-disjointness result for all D-invariant compact sets of Hausdorff dimension < 1/2 (its Theorem 6) via a reduction to SFTs and a difference-set measure-zero lemma, and a local result near the doubling map (its Theorem 7) asserting no joint invariant compact sets exist; these align with the candidate’s two targets. However, Theorem 7 in the paper is misstated: as written it quantifies over all K in KD, which would include K = [0,1) (and ∅), making the claim false; its proof explicitly recognizes these exceptions, so the intended statement should exclude the trivial sets, matching the candidate’s clarification (the paper states the theorem but immediately notes that equality can hold only for the trivial sets under irrational rotation; see the theorem and its proof: Theorem 7 and its discussion ). Conversely, the candidate’s proof of density for a fixed K uses an unjustified inequality on Hausdorff dimensions of sum/difference sets; the paper instead uses an upper box-dimension lemma (Proposition 1) to get measure-zero difference sets, which is sufficient to find arbitrarily small translations and establish density for SFTs, then pass to all K via SFT approximation (Proposition 3, Corollary 1, Theorem 6 ). Thus, the paper has a minor statement-level oversight in Theorem 7, while the model’s argument contains a substantive gap in its use of dimension sums; with the standard box-dimension lemma (as the paper uses) the model’s approach can be repaired.