INVARIANT DISTRIBUTIONS OF PARTIALLY HYPERBOLIC SYSTEMS: FRACTAL GRAPHS, EXCESSIVE REGULARITY, AND RIGIDITY

The paper’s quantitative non-fractal invariance principle (Theorem 3.1) is clearly stated and proved via an obstruction function, a distortion lemma (Lemma 5.1), a minimality dichotomy for the obstruction (Lemma 5.3), and a uniform local oscillation criterion implying a box-dimension lower bound (Lemma 5.5), culminating in Proposition 5.4 (part (1)) and Proposition 5.6 (part (2)) . The candidate solution’s Part (1) is conceptually aligned with the paper and essentially correct. However, Part (2) contains a critical counting error: it claims a single long unstable arc intersects on the order of δ^{-dim M} many δ-balls, which is false—an embedded 1-dimensional (or dim Eu-dimensional) plaque can intersect only O(δ^{-dim Eu}) such balls. The paper avoids this pitfall by deriving a uniform-in-x lower bound on local oscillations h_{α,ε}(x), which directly yields the box-dimension bound dim_B Graph(Φ) ≥ dim M + 1 − A via Lemma 5.5, without any global equidistribution or packing argument .