Center foliation: absolute continuity, disintegration and rigidity

Abstract

In this paper we address the issues of absolute continuity for the center foliation (as well as the disintegration on the non-absolute continuous case) and rigidity of volume preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov on $\mathbb T^3$. It is shown that the disintegration of volume on center leaves may be neither atomic nor Lebesgue. It is also obtained results concerning the atomic disintegration. Moreover, the absolute continuity of the center foliation does not imply smooth conjugacy with its linearization. Imposing stronger conditions besides absolute continuity on the center foliation, smooth conjugacy is obtained

Lyapunov exponents and smooth invariant foliations for partially hyperbolic diffeomorphisms on

“…The first one is because our proof is cleaner than the one made in [7]. In the present paper, Theorem 2.2 is proved without the use of disintegration theory.…”

“…See [7] for an example of a DA diffeomorphism with C 1 centre leaves, but which is not C 1 conjugate to its linearization, compare to Corollary 2.4 below. If we exclude the C 1 hypothesis from Theorem 2.2, then from [8] the Lyapunov exponents are constant volume almost everywhere and equal to the Lyapunov exponent of the linearization.…”

“…Since f is volume preserving the sum of all the Lyapunov exponent of periodic points are zero. Hence, the Lyapunov exponent on each direction is constant on periodic points, by [7,Proposition 1.1] this implies that f is C 1 -conjugate to its linearization.…”

“…The next result has already been proved in [7], but we state and include the proof here for two reasons. The first one is because our proof is cleaner than the one made in [7].…”

“…We prove that the Lyapunov exponents (all three directions) of the periodic points are constant, hence by [7, Proposition 1.1] f is C 1 conjugate to its linearization. In fact, it is C 1 + θ because the proof of Proposition 1.1 from [7] is to reduce the case to use see data periodic property and then use [18] which proves not only C 1 but C 1 + θ .…”