Rigidity of u-Gibbs measures near conservative Anosov diffeomorphisms on T³ (with Sébastien Alvarez, Martin Leguil and Davi Obata) [Vídeo of a Talk at the Brasilian School of Dynamical Systems 2021] This paper reports a research we have ben conducting during the last two years and whose primary goal is to apply in smooth dynamics measure rigidity techniques from homogeneous and Teichmüller dynamics, notably the exponential drift and the facthorization techniques, from Benoist-Quint and Eskin-Mirzakhani, respectively. Here we show that for every C² Anosov diffeomorphism on T³ having a sppliting Es+Ec+Eu, with the center being expanding, if the stable holonomies are C¹ then either Es+Eu is integrable of every fully supported u-Gibbs measure is in fact SRB.
The disintegration of measures with periodic repetitive pattern (with Régis Varão) The driving force for this work was try to understand better non-hyperbolic measures for partially hyperbolic diffeomorphisms with 1D center. We address, however, a very modest toy problem in this direction which is to describe the disintegration of the so-called GIKN non-hyperbolic measures, in their original locally constant skew product setting. We introduce a generalization of their construction, which we call measures with periodic repetitive pattern and prove that these measures have atomic disintegration on the fibers. We also study some general ergodic theoretical properties of these measures.
The semicontinuity lemma.Permanent preprint, not intended for publication. In this short note I give a proof of the well known fact that a semi-continuous function has a generic set of continuity points, with only separability of target space being required.
Existence of attractors, homoclinic tangencies and singular hyperbolicity for flows.(with Alexander Arbieto and C.A Rojas) Permanent preprint, not intended for publication. A gap in our proof of Theorem 2.5, which is based in Morales-Pacífico-Pujals, was noticed by Rafael Potrie and Enrique Pujals. The results we obtained were announced later inCrovisier-Yang and Morales . It seems to me, however, that it would be interesting to try to fix our proof using the ideas of Li-Gan-Wen.
Book
Fluxos Estrela (Star Flows).(With Alexander Arbieto and Tatiana Sodero) Published by IMPA as a text for an advanced course in 28º Colóquio Brasileiro de Matemática. In portuguese.
