SRB Measures for $C^{1+\mathrm{Dini}}$ Diffeomorphisms

For $C^{1+\mathrm{Dini}}$ diffeomorphisms, we prove a Ledrappier--Young-type characterization of SRB measures among invariant measures whose supports admit dominated splittings.

When the dominating bundle has only positive Lyapunov exponents, absolute continuity of the conditional measures on the corresponding Pesin unstable manifolds is equivalent to the partial Pesin entropy formula.

As an application, we obtain SRB measures for partially hyperbolic mostly expanding attractors in the $C^{1+\mathrm{Dini}}$ category.

Counterexamples are provided to show that these conclusions fail in the $C^1$ category, even under uniform hyperbolicity.