ON THE HAUSDORFF DIMENSION OF INVARIANT MEASURES OF PIECEWISE SMOOTH CIRCLE HOMEOMORPHISMS

Frank Trujillo

correctmedium confidence

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

Audit review

The paper proves that for a full-measure set of rotation-type combinatorics, P-homeomorphisms with zero mean nonlinearity have invariant measures of Hausdorff dimension zero, via reduction to AIETs and a zero-HD Rokhlin tower criterion (Proposition 3.3), then verifying the criterion using Zorich-accelerated renormalization, Oseledets splittings, and a dichotomy excluding the 2-interval (stable) case. The candidate solution follows the same blueprint: mapping to GIETs/AIETs, invoking Zorich–Oseledets theory to build rigid towers with uniform contraction/expansion on the base, and applying a zero-HD criterion. The only substantive deviation is a stronger-than-needed growth claim for log-slopes; the paper only requires a uniform deviation from 1 along a subsequence. Otherwise, the logical steps and required hypotheses align closely.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper presents a careful and effective adaptation of renormalization and tower techniques to the zero-mean-nonlinearity setting for P-homeomorphisms. The use of Zorich acceleration and Oseledets splittings, together with a crisp dichotomy for AIETs, substantiates the main theorem. Small clarifications on notation and on transferring tower mass via conjugacy would further improve readability.