dynamical systems

SRB Measures for Diffeomorphisms

arXiv:2607.13530

summary

The paper establishes a Ledrappier–Young type criterion for SRB measures of C^{1+\mathrm{Dini}} diffeomorphisms with dominated splittings, linking absolute continuity on unstable manifolds to a partial Pesin entropy formula, and applies this to partially hyperbolic mostly expanding attractors while showing the results do not hold in the C^1 setting.

Abstract

For 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 category. Counterexamples are provided to show that these conclusions fail in the category, even under uniform hyperbolicity.

Topics & keywords

#srb measures#dominated splitting#pesin theory#partially hyperbolic dynamics#entropy formula#c^1+dini regularitySRBLedrappier-Youngdominated splittingPesin unstable manifoldspartial entropy formulamostly expanding attractorC^{1+\mathrm{Dini}} diffeomorphisms