ISOGENIES OF MINIMAL CANTOR SYSTEMS: FROM STURMIAN TO DENJOY AND INTERVAL EXCHANGES

The paper proves that Sturmian subshifts (X_α,σ_α) and (X_β,σ_β) are isogenous if and only if α and β lie in the same PGL2(Q)-orbit (Theorem 6.6) and gives a concrete chain using flow equivalences (classified by PGL2(Z)) and a 2-AI step realizing α↦mα, via a factorization M=UDV with U,V∈SL2(Z), D=diag(m,1) . It also establishes that, between zero-dimensional systems, flow equivalence is induced by return equivalence (Parry–Sullivan) and classifies Sturmian flow equivalence by PGL2(Z) (Theorem 4.3 and Theorem 4.5) . The key 2-AI effect from the m-th power (X_α,σ_α^m) to (X_{mα},σ_{mα}) is explicitly highlighted in the introduction and used later via Corollary 5.21 . The candidate solution mirrors the forward direction precisely (Steps A–C) and correctly identifies the same operations. For the converse, the model argues that any isogeny chain acts on slopes by compositions of unimodular moves and dilations, hence by PGL2(Q); while this matches the conclusion, the paper instead proves the converse using coinvariant groups and states (Corollary 6.5 and Lemma A.2) rather than an a priori factorization of arbitrary isogeny chains into the simple pattern flow–2-AI–flow (indeed, a general factorization question is posed as open, Question 6.18) . Thus the model’s solution is correct in substance but compresses a nontrivial step; the paper’s proof is complete and rigorous.