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In this paper we mainly address the problem of disintegration of Lebesguemeasure and measure of maximal entropy along the central foliation of(conservative) Derived from Anosov (DA) diffeomorphisms. We prove that foraccessible DA diffeomorphisms of $\mathbb{T}^3$, atomic disintegration has thepeculiarity of being mono-atomic (one atom per leaf). We further provide openand non-empty condition for the existence of atomic disintegration. Finally, weprove some new relations between Lyapunov exponents of DA diffeomorphisms andtheir linearization.

Mono-atomic disintegration and Lyapunov exponents for derived from Anosov diffeomorphisms