## Jairo Bochi

Associate Professor
Facultad de Matemáticas, PUC-Chile

“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

I am a mathematician. I completed my undergraduate studies at UFRGS (Porto Alegre, Brazil) in 1996, and my PhD at IMPA (Rio de Janeiro, Brazil) in 2001. I have previsouly had tenured positions on UFRGS and PUC-Rio. I arrived at PUC-Chile in 2014. My research focuses on Dynamical Systems and its relations to Geometry, Linear Algebra, and Control Theory.

 Address: Facultad de Matemáticas Pontificia Universidad Católica de Chile (PUC-Chile) Av. Vicuña Mackenna #4860, Macul Santiago - Chile
 Office: 322 Phone: +56(02)2354-1025 Email:

### Teaching:

 2018/2 Sistemas Dinámicos (MAT2565), Cálculo IV (MAT1640), Precálculo (MAT1000) 2018/1 - 2017/2 Teoría Ergódica (MPG3960), Precálculo (MAT1000) 2017/1 Cálculo I (MAT1610) 2016/2 Cálculo I (MAT1610) 2016/1 Análisis Real (MAT2515), Variable Compleja (MPG3950) 2015/2 Sistemas Dinámicos (MAT2565), Cálculo III (MAT1630) 2015/1 Teoría de Integración (MAT2535) 2014/2 Cálculo II (MAT220E) 2014/1 Geometría Diferencial (MAT2305)

### Research papers:

Title Joint with... “Slides”   Published in ... Link/File
Abundance of counterexamples to the finiteness conjecture in arbitrary dimension (working title) Çağrı Sert In progress...
Extremal norms for fiber bunched cocycles Eduardo Garibaldi
Emergence and complexity of the set of invariant measures (working title) Pierre Berger In progress...
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 /
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 To appear in Journal of the European Mathematical Society
Flexibility of Lyapunov exponents among conservative diffeormophisms (working title) Anatole Katok, Federico Rodriguez Hertz In progress...
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:

• A proof of Denjoy theorem:
• Structures induced by the symmetric difference metric:
• The basic ergodic theorems, yet again:
• Another proof of the spectral radius formula:
• Note on robustness of periodic measures in ergodic optimization (with Yiwei Zhang):
• Perfect matchings in inhomogeneous random bipartite graphs in random environment (with G. Iommi and M. Ponce):
• An ergodic theorem for permanents of oblong matrices (with G. Iommi and M. Ponce):
• 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).

### Students:

Current:

• Sebastián Pavez Molina (MSc). Research project: Ergodic Optimization.

Former:

• 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: (October, 2017)
• MSc dissertation (Jul 17, 2018): Negative curvature, matrix products, and ergodic theory.
• Renato Velozo Ruiz. MSc dissertation (Jul 12, 2018): Characterization of uniform hyperbolicity for fiber-bunched cocycles.
• Paulo N. Orenstein, co-advised by Carlos Tomei. MSc dissertation (Jan 23, 2014): Optimal transport and the Wasserstein metric.
• Cong Zhou. 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. Text:
• Frederico B. Israel. Undergraduate research project (2009): Hilbert projective metric. Visualization software: Windows executable, instructions, Linux source

### Scientific service:

I'm a member of the editorial board of Discrete and Continuous Dynamical Systems - Series A.

I was one of the organizers of the EDAI Dynamical Systems Seminar for a few years.

### Some math-related bookmarks:

My profiles:

 New Trends in Ergodic Theory Colloquium of PUC-Chile math department Santiago Dynamical Systems seminar Mathematical Imagery by Jos Leys Videos of talks in Dynamical Systems (CUNY Einstein Chair Math. Seminar) Feynman's Messenger Lectures on Project Tuva Proceedings of the ICM's 1893-2010 GDZ database AMS Open Math Notes Allen Hatcher's (free!) Algebraic Topology book Mathematical English Usage Dictionary Origin of some words in Mathematics Hyperbolic maze (a game) Square peg problem (applet) Funny MathReviews

### Personal:

 My very best work (joint with Paula Porto) Checking in at Hilbert's My favorite superhero

 Last update: September, 2018.