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

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

Office: 303 McAllister Email:

Teaching:

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)	-		...
Spectrum maximizing products are not generically unique	Piotr Laskawiec		SIAM Journal on Matrix Analysis and Applications, 45 (2024), no. 1, pp. 585-600.	/
The Halász-Székely barycenter	Godofredo Iommi, Mario Ponce		Proceedings of the Edinburgh Mathematical Society, 65 (2022), pp. 881-911.	/
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.
  • Generic rotation sets: Ergodic Theory and Dynamical Systems, 42 (2022), no. 1, pp. 250-262. /
  • MSc dissertation (Jul 15, 2020): Ergodic optimization and rotation sets.
• Eduardo Oregón-Reyes.
  • Properties of sets of isometries of Gromov hyperbolic spaces: Groups, Geometry, and Dynamics, 12 (2018), no. 3, pp. 889-910. /
  • A new inequality about matrix products and a Berger-Wang formula: Journal de l'École polytechnique - Mathématiques, 7 (2020), p. 185-200. /
  • MSc dissertation (Jul 17, 2018): Negative curvature, matrix products, and ergodic theory.
• 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

Scientific service:

• Together with Svetlana Katok, I'm organizing the 33rd Fall meeting of the Semi-annual Workshop in Dynamical Systems and Related Topics (November 3-6, 2022).
• I'm a member of the editorial board of Discrete and Continuous Dynamical Systems - Series A since 2014.

Some math-related bookmarks:

My profiles:            

Anatole Katok Center for Dynamical Systems and Geometry	arxivist: your personal guide to the arXiv	Celebratio Mathematica: people of mathematics
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 1893-2018	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	Erdős Problems

Last update: November, 2023.