Mather’s regions of instability for annulus diffeomorphisms

The uploaded paper proves exactly the statement in question: under f in Diff^{1+ε} preserving orientation and boundary components, if the annulus A is a Birkhoff region of instability (BZI) and the rotation set of a lift has interior, then there exists an f-invariant essential open annulus A* whose frontier meets both boundary components and A* is a Mather region of instability; see Theorem 1 and the accompanying discussion of K_−, K_+ and the annulus between them, along with the role of the rotation set and BZI in the preliminaries and Lemma 1 (Theorem 1 and Lemma 1 are stated and used explicitly in the text) . The proof leverages rotational topological horseshoes (Proposition 1 and Corollary 1) and a dense set of rational rotation numbers realized by hyperbolic periodic saddles (Theorem 2), producing the required connecting dynamics and the intrinsic construction of A* . The candidate model’s solution outlines a different proof strategy: it realizes rationals via Boyland-type periodic orbits, invokes Le Calvez/Jaulent transverse foliations and forcing theory, and then constructs an invariant open annulus containing orbits connecting the two boundary components. While the model’s outline omits many technical steps (e.g., the paper’s intrinsic K_−/K_+ construction and horseshoe machinery), its claims are consistent with the paper’s result and use standard tools cited there. Hence, the paper’s argument is correct and complete for the stated result, and the model presents a broadly correct but higher-level and different proof outline.