The information topology and true laminations for diffeomorphisms

We explore the lamination structure from data supplied by a general measure space $X$ provided with a Borel probability measure $m$. We show that if the data satisfy some typical axioms, then there exists a lamination $\mathcal{L}$ injected in the underlying space $X$whose image fills up the measure $m$. For an arbitrary $C^{1+\alpha}$-diffeomorphism $f$ of a compact Riemannian manifold $M$, we construct the data that naturally possess the properties of the axioms; thus we obtain the stable and unstable laminations $\mathcal{L}^{s/u}$ continuously injected in the stable and unstable partitions $\mathcal{W}^{s/u}$. These laminations intersect at almost every regular point for the measure.