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Jairo Bochi Professor associado Departamento de Matemática, PUC-Rio

Sobre mim = About me:

Cursei graduação (93-96) e mestrado (96-97) na UFRGS, e doutorado (97-01) no IMPA. Sou professor do Departamento de Matemática da PUC-Rio desde 2008. Minha pesquisa se concentra em Sistemas Dinâmicos.

I completed my undergraduate studies at UFRGS (1996), and my PhD at IMPA (2001). I'm a professor at PUC-Rio since 2008. My research focuses on Dynamical Systems.

Curriculum Vitae

Currículo Lattes = Official CV

Endereço = Address: Departamento de Matemática PUC-Rio Rua Marquês de São Vicente, 225 22451-900 Rio de Janeiro, RJ - Brasil

Sala: L849 Telefone: (21) 3527-1756 Email:

Ensino = Teaching:

Research papers:

Here is a commented list:

Title	Joint with...	“Slides”	Published in ...	Link/File
The entropy of Lyapunov-optimizing measures of some matrix cocycles	Michał Rams		Submitted
Continuity properties of the lower spectral radius	Ian D. Morris		Submitted
Robust criteria for the existence of nonhyperbolic measures (working title)	Christian Bonatti, Lorenzo J. Díaz		In progress...
Generic linear cocycles over a minimal base	-		Studia Mathematica, 218 (2013), no. 2, pp. 167-188.	/
Almost reduction and perturbation of matrix cocycles	Andrés Navas		To appear in Annales de l'Institut Henri Poincaré - Analyse Non Linéaire.	/
Denseness of domination (working title)	-		In progress...
Robust vanishing of all Lyapunov exponents for iterated function systems	Christian Bonatti, Lorenzo J. Díaz		Mathematische Zeitschrift 176 (2014), pp. 469-503.	/
Universal regular control for generic semilinear systems	Nicolas Gourmelon		...
A geometric path from zero Lyapunov exponents to rotation cocycles	Andrés Navas		To appear in Ergodic Theory and Dynamical Systems.	/
Perturbation of the Lyapunov spectra of periodic orbits	Christian Bonatti		Proceedings of the London Mathematical Society 105 (2012), no. 1, pp. 1-48.	/
Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms	Artur Avila		Transactions of the American Mathematical Society 364 (2012), no. 6, pp. 2883-2907.	/
Opening gaps in the spectrum of strictly ergodic Schrödinger operators	Artur Avila, David Damanik		Journal of the European Mathematical Society 14 (2012), no. 1, pp. 61-106	/
Nonuniform center bunching and the genericity of ergodicity among \(C^1\) partially hyperbolic symplectomorphisms	Artur Avila, Amie Wilkinson		Annales Scientifiques de l'École Normale Supérieure 42 (2009), no. 6, pp. 931-979.
Some characterizations of domination	Nicolas Gourmelon		Mathematische Zeitschrift 263 (2009), no. 1, pp. 221-231.	/
Uniformly hyperbolic finite-valued \({\rm SL}(2,\Bbb{R})\) cocycles	Artur Avila, Jean-Christophe Yoccoz		Commentarii Mathematici Helvetici 85 (2010), no. 4, pp. 813-884.	/
\(C^1\)-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents	-		Journal of the Institute of Mathematics of Jussieu, 9 (2010), no. 1, pp. 49-93.	/
Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts	Artur Avila, David Damanik		Duke Mathematical Journal 146 (2009), no. 2, pp. 253-280.	/
Cocycles over generic volume preserving dynamics (working title)	Nicolas Gourmelon		In progress...
A uniform dichotomy for generic \({\rm SL}(2,\Bbb{R})\) cocycles over a minimal base	Artur Avila		Bulletin de la Société Mathématique de France 135 (2007), 407--417.
Generic expanding maps without absolutely continuous invariant \(\sigma\)-finite measure	Artur Avila		Mathematical Research Letters 14 (2007), 721-730.
A generic \(C^1\) map has no absolutely continuous invariant probability measure	Artur Avila		Nonlinearity 19 (2006), 2717-2725.	/
Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for \({\rm SL}(2,\Bbb{R})\) cocycles	Bassam Fayad		Bulletin of the Brazilian Mathematical Society 37 (2003), no. 3, 307-349.	/
A remark on conservative diffeomorphisms	Bassam Fayad, Enrique Pujals		Comptes Rendus Acad. Sci. Paris, Ser. I 342 (2006), 763-766.	/
\(L^p\)-generic cocycles have one-point Lyapunov spectrum	Alexander Arbieto		Stochastics and Dynamics 3 (2003), 73-81. Corrigendum. ibid, 3 (2003), 419-420.	/ +
Lyapunov exponents: How frequently are dynamical systems hyperbolic?	Marcelo Viana		Modern dynamical systems and applications, 271-297, Brin, Hasselblatt, Pesin (eds.) Cambridge Univ. Press, 2004.
Inequalities for numerical invariants of sets of matrices	-		Linear Algebra and its Applications, 368 (2003), 71-81.	/
The Lyapunov exponents of generic volume preserving and symplectic maps	Marcelo Viana		Annals of Mathematics, 161 (2005), No. 3, 1423--1485.
Robust transitivity and topological mixing for \(C^1\)-flows	Flavio Abdenur, Artur Avila		Proceedings of American Mathematical Society, 132 (2004), 699-705.	/
Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps	Marcelo Viana		Annales de l'Institut Henri Poincaré - Analyse non linéaire, 19 (2002), 113-123.	/
A formula with some applications to the theory of Lyapunov exponents	Artur Avila		Israel Journal of Mathematics, 131 (2002), 125-137.	/
Genericity of zero Lyapunov exponents	-		Ergodic Theory and Dynamical Systems, 22 (2002), 1667-1696.	/ ,
Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles	-		Permanent preprint	,

My profile on MathSciNet

My profile on Google Scholar

Other texts: Lecture notes, other talks etc:

• Bochi-Fayad problem:
• Note on the dimension of certain algebraic sets of matrices (with N. Gourmelon):
• Geometria muito além dos gregos (palestra para os calouros):
• Médias aritméticas disfarçadas (palestra Oktobermat):
• Proof of the subadditive ergodic theorem (with A. Avila):
• Notes of the course on Lyapunov exponents given at the Workshop on D.S. at Trieste, with A. Avila: First week:; second week: (manuscript by Aline Cerqueira).
• Notes on a theorem of Furstenberg on random products of matrices:
• Proofs of Oseledets theorem:
  • In dimension 2 via hyperbolic geometry:
  • In any dimension: . Disclaimer: This text is sketchy and has never been revised. David Mesquita's dissertation (in portuguese) contains a more readable proof.
  • Karlsson-Margulis Theorem: (talk slides, in portuguese)
• Dois espaços compactos contráteis com um ponto em comum cuja união não é contrátil (seguindo Elon L. Lima):
• Raízes de polinômios com coeficientes inteiros limitados:

Alunos = Students:

Prévios = Former:

• Paulo N. Orenstein (mestrado; bolsa Faperj Nota 10), em co-orientação com Carlos Tomei. Dissertação (defendida em 23/01/2014): Optimal transport and the Wasserstein metric.
• Cong Zhou (mestrado; bolsa CAPES). Dissertação (defendida em 11/03/2013): Teorema ergódico multiplicativo em espacos métricos de curvatura não-positiva.
• Miguel K. Schnoor (doutorado; bolsa CNPq). Tese (defendida em 03/08/2012): The non-existence of absolutely continuous invariant probabilities is \(C^1\)-generic for flows.
• Leonardo Lobo C. da Fontoura, Frederico B. Israel (iniciação científica; bolsa CNPq). Projeto (concluído em 2011): A geometria dos rastros de bicicleta.
• Pedro Henrique Milet P. Pereira (mestrado; bolsa Faperj Nota 10). Dissertação (defendida em 25/03/2011): Curvas de Peano e campos de direções.
• Paulo N. Orenstein, Frederico B. Israel (iniciação científica). Projeto (concluído em 2009): A métrica projetiva de Hilbert. Texto (de Paulo N. Orenstein) ; programa para visualizar (de Frederico Israel): executável Windows, instruções, fonte Linux.

Some math-related bookmarks:

EDAÍ Dynamical Systems Seminar	DinamiCarioca	Oktobermat
Mathematical Imagery by Jos Leys	Videos of talks in Dynamical Systems (CUNY Einstein Chair Math. Seminar)	Feynman's Messenger Lectures on Project Tuva
GDZ database	Mathematical English Usage - Online Dictionary	Origin of some words in Mathematics
Allen Hatcher's (free!) Algebraic Topology book

Personal stuff:

Last update: January, 2014.