Statistics of torus piecewise isometries

Michael Blank

correctmedium confidence

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

Audit review

The paper’s main theorem (existence of an invariant probability for extendable torus PIs with a finite special partition, and non-atomicity when Γ ∩ Per(T) = ∅) is supported by a coherent approximation scheme via weakly periodic maps and a careful passage to the limit that controls discontinuities and boundary effects. In contrast, the model’s Banach-limit construction does not rigorously establish T-invariance for a discontinuous map (the key step L(φ∘T)=L(φ) is used outside the domain C(X)), and it relies on an incorrect geometric claim that Fix(T^n) is finite, which fails for torus isometries (e.g., reflections have one-dimensional fixed sets).

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript addresses a central open-looking issue for multidimensional piecewise isometries by producing a robust existence result for invariant measures under natural extendability and finiteness assumptions, and by giving a clear non-atomicity criterion tied to the boundary/periodic-point interaction. The approximation by weakly periodic maps, together with Lemmas 10–13, convincingly overcomes the incompatibility of classical Krylov–Bogolyubov with discontinuities. Some expository clarifications (especially regarding the non-atomicity mechanism and the role of periodic points away from Γ) would further strengthen the paper.