## 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

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

“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

### 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.

 Address: Department of Mathematics 109 McAllister Building The Pennsylvania State University University Park, PA 16802, USA
 Office: 227 McAllister Email:

### Teaching:

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
The Halász-Székely barycenter Godofredo Iommi, Mario Ponce Proceedings of the Edinburgh Mathematical Society. /
Flexibility of Lyapunov exponents Anatole Katok, Federico Rodriguez Hertz Ergodic Theory and Dynamical Systems, 42 (2022), pp. 554-591. /
On emergence and complexity of ergodic decompositions Pierre Berger Advances in Mathematics, 390 (2021), 107904. /
Extremal norms for fiber bunched cocycles Eduardo Garibaldi Journal de l'École polytechnique - Mathématiques, 6 (2019), pp. 947-1004. /
Ergodic optimization of Birkhoff averages and Lyapunov exponents - Proceedings of the International Congress of Mathematicians 2018, Rio de Janeiro, vol. 2, pp. 1821-1842. / /
Equilibrium states of generalised singular value potentials and applications to affine iterated function systems Ian D. Morris Geometric and Functional Analysis, 28 (2018), no. 4, pp. 995-1028. /
Dominated Pesin theory: convex sum of hyperbolic measures Christian Bonatti, Katrin Gelfert Israel Journal of Mathematics, 226 (2018), no. 1, pp. 387-417. /
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. /
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. /
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. /
Anosov representations and dominated splittings Rafael Potrie, Andrés Sambarino Journal of the European Mathematical Society 21 (2019), no. 11, pp. 3343-3414. /
Robust criterion for the existence of nonhyperbolic measures Christian Bonatti, Lorenzo J. Díaz Communications in Mathematical Physics 344 (2016), no. 3, pp. 751-795. /
The scaling mean and a law of large permanents Godofredo Iommi, Mario Ponce Advances in Mathematics 292 (2016), pp. 374-409. /
Ergodic optimization of prevalent super-continuous functions Yiwei Zhang International Mathematics Research Notices 2016 (2016), no. 19, pp. 5988-6017. /
Cocycles of isometries and denseness of domination - Quarterly Journal of Mathematics 66 (2015), no. 3, pp. 773-798. /
Peano curves with smooth footprints Pedro H. Milet Monatshefte für Mathematik 180 (2016), no. 4, pp. 693-712. /
The entropy of Lyapunov-optimizing measures of some matrix cocycles Michał Rams Journal of Modern Dynamics 10 (2016), pp. 255-286. /
Continuity properties of the lower spectral radius Ian D. Morris Proceedings of the London Mathematical Society 110 (2015), pp. 477-509. /
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 Annales de l'Institut Henri Poincaré - analyse non linéaire 31 (2014), no. 6, pp. 1101-1107. /
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 Mathematics of Control, Signals, and Systems 26 (2014), no. 4, pp. 481-518. /
A geometric path from zero Lyapunov exponents to rotation cocycles Andrés Navas Ergodic Theory and Dynamical Systems 35 (2015), no. 2, pp. 374-402. /
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. / Correction:
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. /
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), no. 5, 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 (2006), 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. Correction:
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 ,

### Notes and other texts:

• Perfect matchings in inhomogeneous random bipartite graphs in random environment (with G. Iommi and M. Ponce): CUBO 24 (2022) no. 2, pp. 263-272. /
• Flexibility of the entropy of area-preserving Anosov diffeomorphisms (after Anatole Katok): blog
• Genericity of periodic maximization: proof of Contreras' theorem following Huang, Lian, Ma, Xu, and Zhang:
• A proof of Denjoy theorem:
• Structures induced by the symmetric difference metric:
• The basic ergodic theorems, yet again: CUBO 20 (2018) no. 3, pp. 85-95. /
• Another proof of the spectral radius formula:
• Note on robustness of periodic measures in ergodic optimization (with Yiwei Zhang):
• An ergodic theorem for permanents of oblong matrices (with G. Iommi and M. Ponce):
• Bochi-Fayad problem:
• Note on the dimension of certain algebraic sets of matrices (with N. Gourmelon):
• Disguised arithmetic means: (talk slides, in Portuguese).
• Proof of the subadditive ergodic theorem (with A. Avila):
• Notes of the course on Lyapunov exponents given at the Workshop on Dyn. Sys. 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)
• Two contractible compact spaces with a point in common whose union is not contractible (following Elon L. Lima): (in Portuguese).
• Roots of polynomials with bounded integer coefficients: (in Portuguese).

### Former students:

• Sebastián Pavez-Molina.
• Eduardo Oregón-Reyes.
• Renato Velozo Ruiz.
• Characterization of uniform hyperbolicity for fiber-bunched cocycles. Dynamical Systems 35 (2020), no. 1, pp. 124-139. /
• MSc dissertation (Jul 12, 2018):
• Paulo N. Orenstein, co-advised by Carlos Tomei. MSc dissertation (Jan 23, 2014): Optimal transport and the Wasserstein metric.
• Zhou Cong. MSc dissertation (Mar 11, 2013): Multiplicative ergodic theorem in non-positively curved spaces.
• Miguel K. Schnoor. PhD thesis (Aug 3, 2012): The non-existence of absolutely continuous invariant probabilities is $C^1$-generic for flows.
• Pedro Henrique Milet P. Pereira. MSc dissertation (Mar 25, 2011): Peano curves and line fields
• Paulo N. Orenstein. Undergraduate research project (2009): Hilbert projective metric.
• Frederico B. Israel. Undergraduate research project (2009): Hilbert projective metric. Visualization software: Windows executable, instructions, Linux source

### Some math-related bookmarks:

My profiles:

 Anatole Katok Center for Dynamical Systems and Geometry Videos of talks in Dynamical Systems (CUNY Einstein Chair Math. Seminar) Mathematical Imagery by Jos Leys Feynman's Messenger Lectures on Project Tuva Proceedings of the ICM's 1893-2014 GDZ database AMS Open Math Notes The OEIS movie Mathematical English Usage Dictionary Origin of some words in Mathematics Hyperbolic maze (a game) Square peg problem (applet) Funny MathReviews Interactive intro to Fourier transforms arxivist: your personal guide to the arXiv Celebratio Mathematica: people of mathematics

 Last update: October, 2022.