Conditions for Atomic Disintegration to be Monoatomic

In this work, we consider conditions in which atomic disintegration is in fact a monoatomic disintegration. One of the motivations of this work comes from the work of Ponce et al. (J Mod Dyn, 8(1):93-107, 2014); they prove that there is a minimal foliation and a set of full volume which intersects each leaf in one point, but their argument uses some of the hyperbolic structure of the system. We generalize some of their techniques in which we eliminate the need for Markov partitions, which are structures inherited from Anosov diffeomorphisms. We prove some results on which atomic disintegration and some contraction hypothesis on the foliation implies monoatomic disintegration.