PARAMETRIC TOPOLOGICAL ENTROPY ON ORBITS OF ARBITRARY MULTIVALUED MAPS IN COMPACT HAUSDORFF SPACES

The paper proves that the Bowen–Dinaburg–Hood (separated/spanning) entropy hKTRT equals the open-cover (Adler–Konheim–McAndrew type) entropy hAKEK for nonautonomous multivalued maps on compact Hausdorff spaces (Theorem 3.21). The proof uses: (i) for any p in the compact uniformity and ε>0, the cover by p-balls Bp(·,ε/2) to show hsep ≤ hAKEK; and (ii) the uniform Lebesgue-number lemma (Lemma 2.3) to find p,δ so that ball-covers refine any given open cover, then a δ/3–spanning set and δ/2–balls to show hAKEK ≤ hspan; combined with hsep = hspan (Theorem 3.10 via Lemma 3.8), this yields hKTRT = hAKEK. All these steps appear explicitly in the PDF . The candidate solution reproduces the same strategy and technical ingredients. The only minor difference is the handling of strict vs non-strict inequalities in the “ball-cover” step: the paper introduces a δ/3 vs δ/2 buffer to guarantee an open cover; the candidate uses δ throughout. This is easily fixed by the same δ/3–δ/2 fudge and does not affect the asymptotic entropy. Hence both are correct, and the proofs are substantially the same.