RELATIVE UNIFORMLY POSITIVE ENTROPY OF INDUCED AMENABLE GROUP ACTIONS

The paper cleanly proves the four-way equivalence in Theorem 1.1—(1) π rel.-u.p.e.; (2) some n with π̃n rel.-u.p.e.; (3) every n with π̃n rel.-u.p.e.; (4) π̃ rel.-u.p.e.—when Y is fully supported, by first establishing (1)⇔(2)⇔(3) (Theorem 3.5) using the independence-set characterization and product stability, and then (1)⇔(4) via Theorems 4.1–4.2 (with the support hypothesis needed only for (1)⇒(4)) . By contrast, the model’s outline contains a critical gap in the claimed implication (4)⇒(3): it asserts one can extend an open cover from Mn(X) to M(X) while preserving non-denseness on the relevant fiber, but provides no justification; controlling closures under such extensions is delicate and the claim is not valid as stated. The paper avoids this step entirely by a different route. The model also has minor inaccuracies (reversed inequality sign in a covering-number comparison and using Δ instead of e_n), though these do not affect the intended direction for (2)⇒(1). Overall, the paper’s arguments are correct and complete, whereas the model’s proof is incomplete for the (4)⇒(3) direction and thus does not establish the full equivalence.