Center Lyapunov exponents in partially hyperbolic dynamics
Andrey Gogolev Ali Tahzibi
Journal of Modern Dynamics 2014, 8(3&4): 549-576 doi: 10.3934/jmd.2014.8.549
In this survey, we discuss the problem of removing zero Lyapunov exponents of smooth invariant measures along the center direction of a partially hyperbolic diffeomorphism and various related questions. In particular, we discuss disintegration of a smooth invariant measure along the center foliation. We also simplify the proofs of some known results and include new questions and conjectures.
keywords: Center Lyapunov exponent Pesin manifold. partially hyperbolic
Physical measures at the boundary of hyperbolic maps
Vítor Araújo Ali Tahzibi
Discrete & Continuous Dynamical Systems - A 2008, 20(4): 849-876 doi: 10.3934/dcds.2008.20.849
We consider diffeomorphisms of a compact manifold with a dominated splitting which is hyperbolic except for a "small" subset of points (Hausdorff dimension smaller than one, e.g. a denumerable subset) and prove the existence of physical measures and their stochastic stability. The physical measures are obtained as zero-noise limits which are shown to satisfy the Entropy Formula.
keywords: physical measures Dominated splitting equilibrium states partial hyperbolicity stochastic stability. random perturbations
Minimal yet measurable foliations
Gabriel Ponce Ali Tahzibi Régis Varão
Journal of Modern Dynamics 2014, 8(1): 93-107 doi: 10.3934/jmd.2014.8.93
In this paper we mainly address the problem of disintegration of Lebesgue measure along the central foliation of volume-preserving diffeomorphisms isotopic to hyperbolic automorphisms of 3-torus. We prove that atomic disintegration of the Lebesgue measure (ergodic case) along the central foliation has the peculiarity of being mono-atomic (one atom per leaf). This implies the measurability of the central foliation. As a corollary we provide open and nonempty subset of partially hyperbolic diffeomorphisms with minimal yet measurable central foliation.
keywords: disintegration minimal foliations Partial hyperbolicity measurable partitions Lyapunov exponents. derived from Anosov
Daniel Smania Ali Tahzibi Marcelo Viana
Discrete & Continuous Dynamical Systems - A 2007, 17(2): i-ii doi: 10.3934/dcds.2007.17.2i
This special issue of DCDS is dedicated to Carlos Gutierrez and Marco Antonio Teixeira, on the occasion of their 60th birthday.
    Born in Peru, C. Gutierrez obtained his Ph.D. degree from IMPA in 1974, under the supervision of Jorge Sotomayor. In the same year he began to serve as a researcher there. He retired from IMPA and now is a full professor at ICMC-University of São Paulo (USP), where his leadership has been crucial to the development of the research on dynamical systems.

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