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Dynamics of Foliations and Rigidity of Ergodic Measures


Given a regular foliation of a Riemannian manifold, we may ask how complex is the foliation. Among the tools introduced to quantify the complexity of a foliation we have the transverse Laypunov exponents and the geometric entropy. Both tools were shown to be extremely importante to capture dynamical informations of a foliation, although, there is still a huge variety of open questions on this field. The main goal of this project is to make use of recent dynamical tools to better understand the dynamical behavior of a generic foliation. To this end we will study:1. which conditions are sufficient to imply that the transversal Lyapunov splitting is dominated;2. under what conditions the transverse invariant measures are absolutely continuous with respect to the Lebesgue measure on the transversals;3. how can we perturb a foliation with zero geometric entropy to obtain a foliation with positive geometric entropy (if it is possible) and how can we remove zero Lyapunov transverse exponents. We holpe that once developed, the tools cited above comes to be of great importance in the study of the dynamics of a foliation. (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. Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets. Nonlinearity, v. 32, n. 7, p. 2337-2351, JUL 2019. Web of Science Citations: 0.
PONCE, GABRIEL. GENERATING POSITIVE GEOMETRIC ENTROPY FROM RECURRENT LEAVES. Proceedings of the American Mathematical Society, v. 146, n. 10, p. 4389-4404, OCT 2018. Web of Science Citations: 0.
PONCE, G.; TAHZIBI, A.; VARAO, R. On the Bernoulli property for certain partially hyperbolic diffeomorphisms. ADVANCES IN MATHEMATICS, v. 329, p. 329-360, APR 30 2018. Web of Science Citations: 1.
NOVAES, DOUGLAS D.; PONCE, GABRIEL; VARAO, REGIS. Chaos Induced by Sliding Phenomena in Filippov Systems. Journal of Dynamics and Differential Equations, v. 29, n. 4, p. 1569-1583, DEC 2017. Web of Science Citations: 1.

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