Research

Accepted or Published Papers

Rigidity of u-Gibbs measures near conservative Anosov diffeomorphisms in T3  (with Sébastien Alvarez, Martin Leguil and Davi Obata). Journal of the European Mathematical Society – JEMS – 2024 Online First

Dirac physical measures on saddle-type fixed points. (with Pablo Guarino and Pierre-Antoine Guihéneuf). J. Dynam. Differential Equations 34 (2022), no. 2, 983–1048. [Slides, in french] [Journal version on-line]

Existence of common zeros for commuting vector fields on 3-manifolds II. Solving global difficulties (with Sébastien Alvarez and Christian Bonatti). Proc. Lond. Math. Soc. (3) 121 (2020), no. 4, 828-875.

On the centralizer of vector fields: criteria of triviality and genericity results (with Martin Leguil and Davi Obata). Math. Z. 297 (2021), no. 1-2, 283–337 [Jornal version on-line]

Weak* and entropy approximations of nonhyperbolic ergodic measures: a geometrical approach (with Lorenzo Díaz and Katrin Gelfert). Math. Proc. Cambridge Philos. Soc. 169 (2020), no. 3, 507–545.

Dirac physical measures for generic diffeomorphisms. Dynamical Systems, 2018, Vol 33, Nº 02, 185-194.

Existence of common zeros for commuting vector fields on 3-manifolds. (with Christian Bonatti) Annales de L’institut Fourier, 67, nº 04 (2017) p. 1741-1781 

On weakly hyperbolic iterated function systems. (with Alexander Arbieto and André Junqueira) Bul Braz Math Soc, New Series (2017) 48, nº1, p.111-140.

On Araujo’s theorem for flows. (with Alexander Arbieto and Carlos Morales) J Dyn Control Syst 22 (2016) no.1 22-55.

Mixing-like properties for some generic and robust dynamics. (with Alexander Arbieto and Thiago Catalan) Nonlinearity 28 (2015) no. 11 4103-4115.

Lyapunov Stability and Sectional Hyperbolicity for Higher Dimensional Flows. (with Alexander Arbieto and Carlos Morales) Math. Ann. 361 (2015), no. 1-2, 67–75.

Preprints

U-Gibbs measure rigidity for partially hyperbolic endomorphisms on surfaces (with Marisa Cantarino) The goal of this paper is to generalize the results of  my previous paper to the non-invertible setting. Moreover, in this paper we work with non-uniform expansion along the center which provides new and interesting applications. Our result provides a different approach, under different assumptions, to Tsujii’s result on existence of absolutely continuous invariant probability measures for surface partially hyperbolic endomorphisms.

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 in Crovisier-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.   

Talks

Brasilian School of Dynamical Systems 2021

GAR Conference 2024 at IHP

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.

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Thesis

Commuting vector fields and generic dynamics. Phd Thesis

Hiperbolicidade Essencial em Superfícies. Master Thesis (In portuguese)

Notes

Rigidez em teoria ergódica e aplicações em teoria dos números. Slides for a talk to (motivated) undergraduate students. I explain how to deduce Margulis theorem (which solves the Oppenheim Conjecture)

Uma introdução ao argumento de deriva polinomial. In this short note, written in Portuguese, I give a complete detailed proof of Ratner’s theorem in the SL2(R) case.

Kochergin’s Theorem (Smooth special flows over irrational rotations are never mixing). Written in Portuguese for (highly) motivated undergraduate students.

Measurable partitions. Notes for a talk at the Workshop for young researchers-Groups acting on manifolds, Teresópolis, June 2016. 

Ergodic Properties of Generic Diffeomorphisms. Poster – Workshop on Dynamical Systems, Salvador 2011.

Weakly Hyperbolic Iterated Function Systems on Compact Spaces. Poster – II Brazilian School of Dynamical Systems, São Carlos 2012.

Aproximation of omega-limit sets by periodic orbits.

Prova do teorema ergódico de Birkhoff.

Ergodic Decomposition.

Non Explosion of homoclinic classes.

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