Jairo Bochi

Professor of Mathematics

“It is, first and foremost, characteristic of human beings to seek and probe for the truth. And so when we are free from mandatory duties and concerns we are keen to see, hear, and learn things and we think that knowledge of arcane or wondrous things is indispensable for the happy life. From this we can conclude that whatever is true, simple, and pure is most suited to human nature.” – Cicero, De Officiis (translation by Brad Inwood)

“Who but a barbarian could fail to believe that a man cannot stand alone if he wishes to create, that tradition is actually the precondition of creation, not its antithesis?” – Theodore Dalrymple

“Mathematics knows no races or geographic boundaries; for Mathematics, the cultural world is one country.” – David Hilbert

“In Mathematics, unlike elsewhere, wrong notions die off easily. Our capacity for understanding is hampered, foremost, by the inability to dispel false concepts.” – Alexander Beilinson

“The beauty of mathematics lies in uncovering the hidden simplicity and complexity that coexist in the rigid logical framework that the subject imposes.” – David Ruelle

“La Mathématique est l’art de donner le même nom à des choses différentes.” – Henri Poincaré

About me:

My research focuses on Dynamical Systems and its relations to Geometry, Linear Algebra, and Control Theory. I completed my PhD at IMPA (Brazil) in 2001. I have previsouly worked at UFRGS (Brazil), PUC-Rio (Brazil), and PUC-Chile. I came to Penn State in 2021. I'm a member of the Anatole Katok Center for Dynamical Systems and Geometry.

Curriculum Vitae pdf

Department of Mathematics
109 McAllister Building
The Pennsylvania State University
University Park, PA 16802, USA
Office: 303 McAllister
Email: email


Spring 2025: Ergodic Theory (Math 506)

Fall 2024: Real Analysis (Math 501)

Spring 2024: Classical Analysis II (Math 404)

Fall 2023: Real Analysis (Math 501)

Spring 2023: Complex Analysis (Math 502)

Fall 2022: Honors Concepts of Discrete Mathematics (Math 311M)

Spring 2022: Classical Analysis I (Math 403)

Fall 2021: Dynamical Systems II (Math 508)

Research papers:

Title Joint with... “Slides”   Published in ... Link/File
Hypergeometric means and their completion (working title) - pdf ... work
Spectrum maximizing products are not generically unique Piotr Laskawiec SIAM Journal on Matrix Analysis and Applications, 45 (2024), no. 1, pp. 585-600. doi link / arxiv
The Halász-Székely barycenter Godofredo Iommi, Mario Ponce Proceedings of the Edinburgh Mathematical Society, 65 (2022), pp. 881-911. doi link / arxiv
Flexibility of Lyapunov exponents Anatole Katok, Federico Rodriguez Hertz pdf Ergodic Theory and Dynamical Systems, 42 (2022), pp. 554-591. doi link / arxiv
On emergence and complexity of ergodic decompositions Pierre Berger pdf Advances in Mathematics, 390 (2021), 107904. doi link / arxiv
Extremal norms for fiber bunched cocycles Eduardo Garibaldi pdf Journal de l'École polytechnique - Mathématiques, 6 (2019), pp. 947-1004. doi link / arxiv
Ergodic optimization of Birkhoff averages and Lyapunov exponents - pdf Proceedings of the International Congress of Mathematicians 2018, Rio de Janeiro, vol. 2, pp. 1821-1842. link / doi link / arxiv
Equilibrium states of generalised singular value potentials and applications to affine iterated function systems Ian D. Morris pdf Geometric and Functional Analysis, 28 (2018), no. 4, pp. 995-1028. doi link / arxiv
Dominated Pesin theory: convex sum of hyperbolic measures Christian Bonatti, Katrin Gelfert Israel Journal of Mathematics, 226 (2018), no. 1, pp. 387-417. doi link / arxiv
On the approximation of convex bodies by ellipses with respect to the symmetric difference metric - Discrete & Computational Geometry, 60 (2018), no. 4, pp. 938-966. doi link / arxiv
Positivity of the top Lyapunov exponent for cocycles on semisimple Lie groups over hyperbolic bases M. Bessa, M. Cambrainha, C. Matheus, P. Varandas, Disheng Xu Bulletin of the Brazilian Mathematical Society, 49 (2018), no. 1, pp. 73-87. doi link / arxiv
A criterion for zero averages and full support of ergodic measures Christian Bonatti, Lorenzo J. Díaz Moscow Mathematical Journal, 18 (2018), no. 1, pp. 15-61. link / arxiv
Anosov representations and dominated splittings Rafael Potrie, Andrés Sambarino pdf Journal of the European Mathematical Society 21 (2019), no. 11, pp. 3343-3414. doi link / arxiv
Robust criterion for the existence of nonhyperbolic measures Christian Bonatti, Lorenzo J. Díaz Communications in Mathematical Physics 344 (2016), no. 3, pp. 751-795. doi link / arxiv
The scaling mean and a law of large permanents Godofredo Iommi, Mario Ponce Advances in Mathematics 292 (2016), pp. 374-409. doi link / arxiv
Ergodic optimization of prevalent super-continuous functions Yiwei Zhang pdf International Mathematics Research Notices 2016 (2016), no. 19, pp. 5988-6017. doi link / arxiv
Cocycles of isometries and denseness of domination - Quarterly Journal of Mathematics 66 (2015), no. 3, pp. 773-798. doi link / arxiv
Peano curves with smooth footprints Pedro H. Milet Monatshefte für Mathematik 180 (2016), no. 4, pp. 693-712. doi link / arxiv
The entropy of Lyapunov-optimizing measures of some matrix cocycles Michał Rams pdf Journal of Modern Dynamics 10 (2016), pp. 255-286. doi link / arxiv
Continuity properties of the lower spectral radius Ian D. Morris Proceedings of the London Mathematical Society 110 (2015), pp. 477-509. doi link / arxiv
Generic linear cocycles over a minimal base - Studia Mathematica 218 (2013), no. 2, pp. 167-188. doi link / arxiv
Almost reduction and perturbation of matrix cocycles Andrés Navas Annales de l'Institut Henri Poincaré - analyse non linéaire 31 (2014), no. 6, pp. 1101-1107. doi link / arxiv
Robust vanishing of all Lyapunov exponents for iterated function systems Christian Bonatti, Lorenzo J. Díaz Mathematische Zeitschrift 176 (2014), pp. 469-503. doi link / arxiv
Universal regular control for generic semilinear systems Nicolas Gourmelon pdf Mathematics of Control, Signals, and Systems 26 (2014), no. 4, pp. 481-518. doi link / arxiv
A geometric path from zero Lyapunov exponents to rotation cocycles Andrés Navas pdf Ergodic Theory and Dynamical Systems 35 (2015), no. 2, pp. 374-402. doi link / pdf
Perturbation of the Lyapunov spectra of periodic orbits Christian Bonatti pdf Proceedings of the London Mathematical Society 105 (2012), no. 1, pp. 1-48. doi link / pdf
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. doi link / pdf
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. doi link / pdf Correction: pdf
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. link / pdf
Some characterizations of domination Nicolas Gourmelon Mathematische Zeitschrift 263 (2009), no. 1, pp. 221-231. doi link / arxiv
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. doi link / pdf
\(C^1\)-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents - pdf Journal of the Institute of Mathematics of Jussieu, 9 (2010), no. 1, pp. 49-93. doi link / pdf
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. doi link / pdf
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. link / pdf
Generic expanding maps without absolutely continuous invariant \(\sigma\)-finite measure Artur Avila Mathematical Research Letters 14 (2007), no. 5, 721-730. doi link / pdf
A generic \(C^1\) map has no absolutely continuous invariant probability measure Artur Avila Nonlinearity 19 (2006), 2717-2725. doi link / pdf
Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for \({\rm SL}(2,\Bbb{R})\) cocycles Bassam Fayad Bulletin of the Brazilian Mathematical Society 37 (2006), no. 3, 307-349. doi link / pdf
A remark on conservative diffeomorphisms Bassam Fayad, Enrique Pujals Comptes Rendus Acad. Sci. Paris, Ser. I 342 (2006), 763-766. doi link / pdf
\(L^p\)-generic cocycles have one-point Lyapunov spectrum Alexander Arbieto Stochastics and Dynamics 3 (2003), 73-81. Corrigendum. ibid, 3 (2003), 419-420. doi link / pdf + pdf
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. pdf Correction: pdf
Inequalities for numerical invariants of sets of matrices - Linear Algebra and its Applications, 368 (2003), 71-81. doi link / pdf
The Lyapunov exponents of generic volume preserving and symplectic maps Marcelo Viana Annals of Mathematics, 161 (2005), no. 3, 1423-1485. doi link / pdf
Robust transitivity and topological mixing for \(C^1\)-flows Flavio Abdenur, Artur Avila Proceedings of American Mathematical Society, 132 (2004), 699-705. doi link / pdf
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. doi link / pdf
A formula with some applications to the theory of Lyapunov exponents Artur Avila Israel Journal of Mathematics, 131 (2002), 125-137. doi link / pdf
Genericity of zero Lyapunov exponents - Ergodic Theory and Dynamical Systems, 22 (2002), 1667-1696. doi link / ps, pdf
Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles - Permanent preprint ps, pdf

Notes and other texts:

Former students:

Scientific service:

Some math-related bookmarks:

My profiles: AMS MathSciNet   Zentralblatt MATH   ArXiv   Researchgate   Wordpress   Math Genealogy   MathOverflow

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