ENTROPY AND PERIODIC ORBITS FOR ASH ATTRACTORS

The paper proves that every three-dimensional ASH attractor (i) contains a periodic orbit (Theorem 2.2) and (ii) is entropy‑expansive (Theorem 2.5) via a route that first obtains positive entropy and then uses a known periodic‑orbit criterion, and later establishes kinematic expansiveness to deduce entropy‑expansiveness . The candidate solution reaches the same two conclusions but sketches a different proof strategy: Liao‑type closing for quasi‑hyperbolic strings for (i), and a plaque/area‑growth argument at hyperbolic times for (ii). While the model’s outline omits several technical lemmas needed to control behavior near singularities (handled carefully in the paper through adapted cross‑sections, partitions, and kinematic expansiveness), its logic is consistent with established tools for dominated splittings and ASH hyperbolic times (cf. Eq. (2.3) in the paper) . Hence both are correct, using different proof architectures.