QUANTITATIVE STATISTICAL STABILITY FOR THE EQUILIBRIUM STATES OF PIECEWISE PARTIALLY HYPERBOLIC MAPS.

The paper proves the O(δ^ζ log δ) quantitative statistical stability on the anisotropic pair (S∞, L∞) via a bespoke "ζ-uniform family" framework (Definition 4.1) and a perturbative fixed-point estimate (Lemma 4.2), requiring only a strong→weak bound on (L0−Lδ) evaluated at the perturbed invariant state μδ and exponential convergence to equilibrium for L0. Crucially, the paper circumvents any need for a full strong→weak operator-norm bound on Lδ−L0 for all μ∈S∞ by first proving uniform Hölder regularity of the disintegration of μδ (Theorem E) and then estimating ||(F0∗−Fδ∗)μδ||∞≤Cδ^ζ (Theorem 5.3), which suffices to apply their abstract stability theorem B and yields ||μδ−μ0||∞=O(δ^ζ log δ) (Theorem D). The candidate solution, by contrast, invokes Keller–Liverani with the claim that ||Lδ−L0||_{S∞→L∞}=O(δ^ζ) holds as an operator norm and that a uniform spectral gap holds for all δ. In this setting S∞ does not enforce cross-fiber Hölder regularity, so the key “moving fiber” term cannot be controlled for arbitrary μ∈S∞; the paper avoids this by working only with μδ whose disintegration is uniformly Hölder (Theorem E). Hence the model’s proof misses a necessary hypothesis and relies on an unjustified operator-norm perturbation bound.