Open sets of partially hyperbolic skew products having a unique SRB measure

The paper’s Theorem A proves that for N large there is a C2-neighborhood U_sk^N of the skew product f_N on T^4 and a C2-open, C2-dense V such that every g in V of class C^{2+α} has a unique SRB measure that is hyperbolic and Bernoulli, with full Lebesgue basin and full support; this is stated explicitly and proved via a classification of u-Gibbs measures (Theorems B, D) and adaptations of Brown–Rodriguez Hertz, plus quantified Pesin theory and an appendix establishing C2-regularity of unstable holonomies under a (2,α)-center-bunching condition . The construction of V leverages Horita–Sambarino’s results on nontrivial accessibility classes (C1-open, C2-dense) for skew products over a fixed Anosov base, not global accessibility, and the paper notes it is unknown whether f_N is accessible . By contrast, the model’s solution hinges on two unsupported steps in this setting: (i) it assumes a C2-open, C2-dense subfamily of accessible maps (via general stable accessibility density) and (ii) it invokes “pinching-and-twisting” genericity for the center derivative cocycle and Avila–Viana’s invariance principle to force nonzero center exponents for every u-Gibbs measure. The genericity result cited is established for cocycles over hyperbolic bases, not for the partially hyperbolic bases at hand; the paper instead proves nonvanishing center exponents near f_N by adapting Berger–Carrasco’s estimates and related techniques . Moreover, the model’s claim that any hyperbolic u-Gibbs measure is automatically SRB (via Ledrappier–Young and holonomy absolute continuity) omits essential hypotheses; the paper bridges this gap through a detailed entropy/invariance-principle analysis (Tahzibi–Yang) and su-torus alternatives (Theorems 6.3, 8.1, 9.1) . Hence the paper’s argument is correct in scope and assumptions, while the model’s proof leaves critical steps unjustified in this context.