{"id":32,"date":"2017-07-19T10:32:53","date_gmt":"2017-07-19T13:32:53","guid":{"rendered":"http:\/\/www.professores.uff.br\/brunosantiago\/?page_id=32"},"modified":"2026-02-07T10:29:19","modified_gmt":"2026-02-07T13:29:19","slug":"pesquisa","status":"publish","type":"page","link":"https:\/\/www.professores.uff.br\/brunosantiago\/pesquisa\/","title":{"rendered":"Research"},"content":{"rendered":"

Accepted or Published Papers<\/strong><\/span><\/h2>\n
U-Gibbs measure rigidity for partially hyperbolic endomorphisms on surfaces<\/a> (with Marisa Cantarino). Accepted for publication (2026) in Ergodic Theory and Dynamical Systems<\/h5>\n
Rigidity of u-Gibbs measures near conservative Anosov diffeomorphisms in T3<\/a>\u00a0 (<\/span>with S\u00e9bastien Alvarez, Martin Leguil<\/a> and Davi Obata<\/a>). Journal of the European Mathematical Society – JEMS – 2024 Online First<\/h5>\n
Dirac physical measures on saddle-type fixed points.<\/a> (with Pablo Guarino and Pierre-Antoine Guih\u00e9neuf<\/a>). J. Dynam. Differential Equations 34 (2022), no. 2, 983\u20131048.\u00a0<\/em>[Slides<\/a>, in french] [Journal version on-line<\/a>]<\/h5>\n
Existence of common zeros for commuting vector fields on 3-manifolds II. Solving global difficulties<\/a>\u00a0(with S\u00e9bastien Alvarez and Christian Bonatti<\/a>). Proc. Lond. Math. Soc. (3) 121 (2020), no. 4, 828-875.<\/h5>\n
On the centralizer of vector fields: criteria of triviality and genericity results<\/a> (with Martin Leguil<\/a> and Davi Obata<\/a>). Math. Z. 297 (2021), no. 1-2, 283\u2013337\u00a0<\/em>[Jornal version on-line<\/a>]<\/h5>\n
Weak* and entropy approximations of nonhyperbolic ergodic measures: a geometrical approach<\/a> (with Lorenzo D\u00edaz<\/a> and Katrin Gelfert<\/a>). Math. Proc. Cambridge Philos. Soc. 169 (2020), no. 3, 507\u2013545.<\/h5>\n
Dirac physical measures for generic diffeomorphisms.<\/a>\u00a0Dynamical Systems, 2018, Vol 33, N\u00ba 02, 185-194.<\/em><\/h5>\n
Existence of common zeros for commuting vector fields on 3-manifolds.<\/a> <\/em>(with Christian Bonatti<\/a>) Annales de L’institut Fourier, 67, n\u00ba 04 (2017) p. 1741-1781\u00a0<\/em><\/h5>\n

\nOn weakly hyperbolic iterated function systems.<\/a> <\/em>(with Alexander Arbieto<\/a> and Andr\u00e9 Junqueira) Bul\u00a0Braz Math Soc, New Series (2017) 48, n\u00ba1, p.111-140.<\/em><\/h5>\n
On Araujo’s theorem for flows.<\/a> <\/em>(with Alexander Arbieto<\/a> and Carlos Morales) J Dyn Control Syst 22 (2016) no.1 22-55.<\/em><\/h5>\n
Mixing-like properties for some generic and robust dynamics.<\/a> <\/em>(with Alexander Arbieto<\/a> and Thiago Catalan) Nonlinearity 28 (2015) no. 11 4103-4115.<\/em><\/h5>\n
Lyapunov Stability and Sectional Hyperbolicity for Higher Dimensional Flows.<\/a> <\/em>(with Alexander Arbieto<\/a> and Carlos Morales) Math. Ann. 361 (2015), no. 1-2, 67\u201375.<\/em><\/h5>\n

Preprint<\/strong><\/span>s<\/strong><\/span><\/h1>\n
Entropy rigidity of u-Gibbs measures<\/a> (with V\u00edtor Gomes). This paper is motivated by the following question: when a measure of maximal entropy is smooth along the unstable foliation?\u00a0<\/em>The expected answer is: seldomly! Moreover, one would expect that such an equality strongly constraints the dynamical system. In this new preprint we show that, in the partially hyperbolic scenario, this expectation is confirmed, by proving that whenever a u-Gibbs measure admits a Margulis family of unstable measures, the Jacobian periodic data along the unstable bundle are constant. We derive a number of corollaries of this, including the case of smooth jointly integrable Anosov partially hyperbolic diffeomorphisms with expanding center, for which the existence of u-Gibbs measure which is also a mesure of maximal unstable entropy implies smooth conjugacy with a linear model along the two dimensional center unstable foliation.<\/h5>\n
The disintegration of measures with periodic repetitive pattern\u00a0<\/a>(with R\u00e9gis Var\u00e3o) 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.\u00a0 We also study some general ergodic theoretical properties of these measures.<\/h5>\n
\u00a0The semicontinuity lemma.<\/a>\u00a0Permanent 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. \u00a0<\/em><\/h5>\n
Existence of attractors, homoclinic tangencies and singular hyperbolicity for flows.<\/a> <\/em>(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-Pac\u00edfico-Pujals, was noticed by Rafael Potrie and Enrique Pujals. The results we obtained were announced later in<\/em>\u00a0Crovisier-Yang<\/a>\u00a0and\u00a0Morales<\/a> . It seems to me, however, that it would\u00a0be interesting to try to fix our\u00a0proof using the ideas of Li-Gan-Wen<\/a>.\u00a0\u00a0\u00a0<\/em><\/h5>\n

Talks<\/strong><\/span><\/h2>\n
Brasilian School of Dynamical Systems 2021<\/a><\/h5>\n

GAR Conference 2024 at IHP<\/a><\/strong><\/p>\n

Book<\/strong><\/span><\/h2>\n
Fluxos Estrela (Star Flows).<\/a>(With Alexander Arbieto and Tatiana Sodero) Published by IMPA as a text for an advanced course in 28\u00ba Col\u00f3quio Brasileiro de Matem\u00e1tica. In portuguese.<\/h5>\n
\n

Thesis<\/strong><\/span><\/h2>\n
Commuting vector fields and generic dynamics.<\/a>\u00a0Phd Thesis<\/h5>\n
Hiperbolicidade Essencial em Superf\u00edcies.<\/a>\u00a0Master Thesis (In portuguese)<\/h5>\n

Notes<\/strong><\/span><\/h2>\n
Rigidez em teoria erg\u00f3dica e aplica\u00e7\u00f5es em teoria dos n\u00fameros<\/a>. Slides for a talk to (motivated) undergraduate students. I explain how to deduce Margulis theorem (which solves the Oppenheim Conjecture)<\/h5>\n
Uma introdu\u00e7\u00e3o ao argumento de deriva polinomial<\/a>. In this short note, written in Portuguese, I give a complete detailed proof of Ratner’s theorem in the SL2(R) case.<\/h5>\n
Kochergin’s Theorem (Smooth special flows over irrational rotations are never mixing). <\/a>Written in Portuguese for (highly) motivated undergraduate students.<\/h5>\n
Measurable partitions<\/a>. Notes for a talk at the Workshop for young researchers-Groups acting on manifolds, Teres\u00f3polis, June 2016.\u00a0<\/em><\/h5>\n
Ergodic Properties of Generic Diffeomorphisms.<\/a> Poster – Workshop on Dynamical Systems, Salvador 2011.<\/h5>\n
Weakly Hyperbolic Iterated Function Systems on Compact Spaces.<\/a> Poster – II Brazilian School of Dynamical Systems, S\u00e3o Carlos 2012.<\/h5>\n
Aproximation of omega-limit sets by periodic orbits.<\/a><\/h5>\n
Prova do teorema erg\u00f3dico de Birkhoff.<\/a><\/h5>\n
Ergodic Decomposition.<\/a><\/h5>\n
Non Explosion of homoclinic classes.<\/a><\/h5>\n
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Accepted or Published Papers U-Gibbs measure rigidity for partially hyperbolic endomorphisms on surfaces (with Marisa Cantarino). Accepted for publication (2026) in Ergodic Theory and Dynamical Systems Rigidity of u-Gibbs measures near conservative Anosov diffeomorphisms in T3\u00a0 (with S\u00e9bastien Alvarez, Martin Leguil and Davi Obata). Journal of the European Mathematical Society – JEMS – 2024 Online […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"categories":[],"tags":[],"class_list":["post-32","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/pages\/32","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/comments?post=32"}],"version-history":[{"count":60,"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/pages\/32\/revisions"}],"predecessor-version":[{"id":903,"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/pages\/32\/revisions\/903"}],"wp:attachment":[{"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/media?parent=32"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/categories?post=32"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.professores.uff.br\/brunosantiago\/wp-json\/wp\/v2\/tags?post=32"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}