Smoothness of stable holonomies inside center-stable manifolds

Under a suitable bunching condition, we establish that stable holonomies inside center-stable manifolds for C1+β diffeomorphisms are uniformly bi-Lipschitz and, in fact, C1+H ölder . This verifies the ergodicity of suitably center-bunched, essentially accessible, partially hyperbolic C1+β diffeomorphisms and verifies that the Ledrappier–Young entropy formula holds for C1+β diffeomorphisms of compact manifolds.