DENSITY OF THE LEVEL SETS OF THE METRIC MEAN DIMENSION FOR HOMEOMORPHISMS

The paper proves density of H^β_α(N) in Hom(N) for any 0 ≤ α ≤ β ≤ n (n ≥ 2) by (i) explicitly constructing on the n-cube a homeomorphism with prescribed lower/upper metric mean dimensions (Lemma 2.4), and (ii) gluing this local model along a periodic orbit using good charts and a boundary-fixing extension, yielding a C0-small perturbation with the desired metric mean dimensions (Theorem 1.5). These steps are clearly stated: the level-set definition H^β_α(N) is given, and the main density theorem is Theorem 1.5, with the cube realizations developed in §2 (Lemmas 2.3–2.4) and the manifold pasting in §3 via exponential charts and annulus maps H_i (the ‘Good Charts’ and gluing scheme) . The candidate solution follows the same blueprint: build a boundary-fixing cube model with the right liminf/limsup behavior, then paste it inside a small chart so the C0-distance (forward and inverse) to the original map is arbitrarily small, and finally argue that the complement contributes zero metric mean dimension. Differences are mainly stylistic: the candidate speaks of a sequence of “variable-thickness horseshoes” and “return times” to control oscillations, whereas the paper’s cube construction uses disjoint embedded horseshoes with controlled sizes and a two-half cube splitting to realize α < β (Lemma 2.4), and then glues along periodic orbits. The paper also explicitly formalizes the chart-based pasting and the invariance/zero-complexity of the complement (e.g., the statement mdim_M(N\K, ϕ_α|) = 0) within its proof sketch, while the candidate appeals to a global pasting scheme at a high level. Overall, the arguments are substantially the same and consistent with the paper’s results and structure .