Synchronizing Dynamical Systems: Finitely Presented Systems and Ruelle Algebras

The paper’s Theorem 5.6 proves that for a mixing finitely presented system with its minimal u-resolving Smale space extension π, the map π×π is a groupoid isomorphism Gs(X,ϕ,Q) → Glcs(Y,f,P), with the s↔u variant and equivariance, via a careful argument built on Lemmas 5.1, 5.3, and 5.4 and a nontrivial surjectivity step . The candidate solution’s Step 3 hinges on assuming uniform continuity of (π|Xu(Q))−1 on plaques to transfer the lcs property downstairs to stable-equivalence upstairs. The paper explicitly warns this is the obstruction: although (π|Xu(Q))−1 is a homeomorphism, there is no reason it is uniformly continuous; if it were, a simple proof would suffice, but this is precisely what fails in general (see Remark 5.7) . Hence the model’s proof skips a critical technical point that the paper resolves with a different, longer argument. The rest of the model’s outline (disjointness of unstable basins, leafwise bijection/homeomorphism, equivariance, and symmetry) aligns with the framework and definitions used in the paper (e.g., the definition of Glcs on Xu(P) and the role of the minimal u-resolving cover), but the crucial lifting step is unsupported .