  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.
