Tubular Dimension: Leaf-Wise Asymptotic Local Product Structure, and Entropy and Volume Growth

We study the conditional measures of every invariant probability measure for every C1+γ diffeomorphism of a closed Riemannian manifold, and we show that they always admit an asymptotic local product structure. This is a direct application of the notion of tubular dimension which we introduce and study in the manuscript. As an additional application we prove a bound on any two consecutive conditional entropies in terms of volume growth.