Synchronizing Dynamical Systems: Their Groupoids and C*-Algebras

The paper proves, for mixing finitely presented systems, (i) amenability of Glc_sync, Glcs, and Glcu; (ii) the ideal exact sequence for A(X,ϕ) with Isync; (iii) a Morita equivalence Isync ≃ S ⊗ U via a bibundle Z; and (iv) simplicity of S, U, and Isync. These are stated and proved in Theorems 5.8, 5.9, 6.9, and 6.10 (and 6.11 in a special case) of the paper. In contrast, the candidate solution crucially asserts that Glcs and Glcu are AF étale groupoids in general and uses this to deduce amenability and simplicity; the paper does not claim AF in this generality and instead proves amenability by lifting to a Smale space and using Borel amenability arguments (Theorem 6.9), then deduces amenability of Glc_sync via the Morita equivalence (Theorem 6.10). The model also omits the density hypotheses needed for the groupoid equivalence Isync ≃ S ⊗ U (explicit in Theorem 5.9 and ensured for finitely presented systems in Lemma 6.7), and it relies on an unsubstantiated “unique splitting” of local conjugacies on product rectangles. Therefore, while the model’s conclusions overlap with the paper’s main theorems, its key steps and assumptions are incorrect or incomplete relative to the paper’s precise hypotheses and proofs. See Theorem 0.1 and Theorems 5.8, 5.9, 6.9, 6.10 in the paper for the correct statements and routes of proof .