POSITIVE LYAPUNOV EXPONENTS ALONG Ecu WITH A UNIFORM GAP IMPLY NON-UNIFORMLY EXPANDING FOR PARTIALLY HYPERBOLIC SYSTEMS

The paper’s Theorem A is proved by first identifying the essential supremum of limsup 1/n log norms with the supremum of Lyapunov exponents over observable measures under a uniform 1-gap hypothesis (Theorem 4.1), using dominated splitting on the support of a Lyapunov-maximizing observable measure and almost-additivity on a dominated subset; see Theorem A and Theorem 4.1 along with the dominated-splitting and cone-field machinery and the Catsigeras–Enrich observable-measure framework . Given the hypothesis that Lebesgue-a.e. limsup is negative, this forces all observable measures to have negative Lyapunov exponent, and a general subadditive-to-block negativity step (their Theorem 6.1) yields the desired K-block non-uniform expansion for Lebesgue-a.e. point . In contrast, the model’s key Step (3) attempts to lower-bound log‖A^n(x)‖ by a sum of logs taken over disjoint dominated segments along the orbit; this inequality is not valid without additional conorm control across the “gaps,” so the argument has a fatal gap. Replacing Step (3) by the paper’s Theorem 4.1 would fix the proof, but as written the model’s derivation is incorrect.