GENERICITY OF HOMEOMORPHISMS WITH FULL MEAN HAUSDORFF DIMENSION

Jeovanny M. Acevedo

correctmedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves that G_m = {ϕ ∈ Homeo(N) : mdimH(N,d,ϕ) = m} contains a residual subset by explicitly constructing ‘strong horseshoes’ and computing a sharp lower bound mdimH = m on a residual set (Theorem 5.4), relying only on the general inequality mdimH ≤ mdimM and a detailed horseshoe calculus . The model’s solution reaches the same conclusion via a different route: it uses (i) the universal upper bound mdimH ≤ m, (ii) the equality mdimH = mdimM under tame growth of covering numbers (for Riemannian metrics), and (iii) the known genericity result that mdimM = m for C^0-generic homeomorphisms on m-manifolds with m ≥ 2. Both arguments are compatible with the paper’s standing assumption n=m ≥ 2 stated in the abstract .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper establishes a sharp and natural genericity theorem for mean Hausdorff dimension on compact Riemannian manifolds of dimension m ≥ 2. The argument is constructive, clear, and complements known results on metric mean dimension. Minor additions (explicitly citing the ambient upper bound for mdim\_H and supplying the omitted residuality proof or a precise reference) would make the manuscript fully self-contained and even more accessible.