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

We introduce the notion of tubular dimension, and give a formula for it. As an application we show that every invariant measure of a $C^{1+\gamma}$ diffeomorphism of a closed Riemannian manifold admits an asymptotic local product structure for conditional measures on intermediate foliations of unstable leaves. As a second application, we prove a bound on the gap between any two consecutive conditional entropies, in the form of volume growth. As a third application, for certain $C^\infty$ maps we compute all conditional entropies for the measure of maximal entropy; And in particular as a consequence, in a follow-up paper we compute the Hausdorff dimension of the equilibrium measure of holomorphic endomorphisms of $\mathbb{C}\mathbb{P}^k$, $k\geq 1$, giving a solution to the Binder-DeMarco conjecture, and answering a question of Forn{\ae}ss and Sibony.