Régis Varão

Center foliation: absolute continuity, disintegration and rigidity

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 automorphism on

Chaos Induced by Sliding Phenomena in Filippov Systems

\mathbb{T}^{3}

Lyapunov exponents and smooth invariant foliations for partially hyperbolic diffeomorphisms on

. It is shown that the disintegration of volume on center leaves for these diffeomorphisms may be neither atomic nor Lebesgue, in contrast to the dichotomy (Lebesgue or atomic) obtained by Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. Preprint , 2012, arXiv:1110.2365v2 ] for perturbations of time-one of geodesic flow. In the case of atomic disintegration of volume on the center leaves of an Anosov diffeomorphism on

Minimal yet measurable foliations

\mathbb{T}^{3}

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

, we show that it has to be one atom per leaf. Moreover, we show that not even a

Rigidity for partially hyperbolic diffeomorphisms

C^{1}

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

center foliation implies a rigidity result. However, for a volume-preserving partially hyperbolic diffeomorphism isotopic to a linear Anosov automorphism, assuming the center foliation is

Measure Rigidity and Disintegration: Time-one map of flows

C^{1}

Invariant measures for Filippov systems

and transversely absolutely continuous with bounded Jacobians, we obtain smooth conjugacy to its linearization.

Invariant measures for Fillipov systems

In this paper we provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit