Smoothness of stable holonomies inside center-stable manifolds and the $C^2$ hypothesis in Pugh-Shub and Ledrappier-Young theory

Under a suitable bunching condition, we establish that stable holonomies inside center-stable manifolds for $C^{1+\beta}$ diffeomorphisms are uniformly bi-Lipschitz and in fact $C^{1+\text{H\"older}}$. This verifies that the Pugh-Shub theory for ergodicity holds for suitably center-bunched, $C^{1+\beta}$, essentially accessible, partially hyperbolic diffeomorphism and verifies that the Ledrappier-Young entropy formulas hold for $C^{1+\beta}$ diffeomorphisms of compact manifolds.