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Absolute continuity of the central foliation for partially hyperbolic systems

Grant number: 11/21214-3
Support Opportunities:Scholarships in Brazil - Post-Doctorate
Effective date (Start): March 01, 2012
Effective date (End): October 31, 2014
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Ali Tahzibi
Grantee:José Régis Azevedo Varão Filho
Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Associated scholarship(s):12/06553-9 - Partially hyperbolic diffeomorphisms: absolute continuity, rigidity, Lyapunov exponents, BE.EP.PD


We shall discuss on this project the absolute continuity of the central foliation for partially hyperbolic diffeomorphisms. The starting point is the PhD thesis of the candidate which among other results provided the first example on $ \mathbb T^3 $ of a conservative "Derived from Anosov" with disintegration of the volume in the central leaves singular with respect to Lebesgue and non-atomic. We conjecture that all conservative Anosov on $\mathbb T ^ 3$ seen as partially hyperbolic ($ TM = E^s \oplus E^c \oplus E^u $) has non-atomic disintegration, the verification of this conjecture is one of the goals to this post-doctoral project. One result from the thesis of the candidate, showed the relationship between mensurability (in the sense of Rokhlin) and it's disintegration. But some more powerful tools are necessary for better understanding the non-mensurability of the foliation. The articles of Ledrappier, Young are the candidates for the first approach, by treating the case of non-compact foliation (which is not always mensurable) through what they called "subordinate partition". (AU)

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Scientific publications (4)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
PONCE, GABRIEL; TAHZIBI, ALI; VARAO, REGIS. MINIMAL YET MEASURABLE FOLIATIONS. JOURNAL OF MODERN DYNAMICS, v. 8, n. 1, p. 93-107, . (09/16792-8, 12/14620-8, 11/21214-3, 12/06553-9)
VARAO, R.. Lyapunov exponents and smooth invariant foliations for partially hyperbolic diffeomorphisms on. DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL, v. 30, n. 2, p. 189-199, . (12/06553-9, 11/21214-3)
PONCE, G.; TAHZIBI, A.; VARAO, R.. On the Bernoulli property for certain partially hyperbolic diffeomorphisms. ADVANCES IN MATHEMATICS, v. 329, p. 329-360, . (09/16792-8, 15/02731-8, 11/21214-3, 16/05384-0, 14/23485-2, 12/14620-8, 17/06463-3, 16/22475-9)
VARAO, REGIS. Center foliation: absolute continuity, disintegration and rigidity. Ergodic Theory and Dynamical Systems, v. 36, n. 1, p. 256-275, . (11/21214-3, 12/06553-9)

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