Prof. Dr. Artur Avila

Vita

• CV (PDF version)

Current Positions

• Since 2018: Professor at the Institute of Mathematics, University of Zurich
• Since 2009: Extraordinary researcher (Cátedra Armínio Fraga) at IMPA, CNRS UMI 2294

Previous Positions

• From 2001 to 2003: Maître de Conférence Associé at the Collège de France
• From 2003 to 2008: Chargé de recherche at the CNRS
• From 2003 to 2006: Attached to the Laboratoire de Probabilités et Modèles Aléatoires (UMR 7599, CNRS, Université Paris Diderot, UPMC)
• From 2006 to 2008: On leave
• From 2006 to 2009: Research fellow of the Clay Mathematics Institute, attached to IMPA, CNRS UMI 2294
• Since 2008: Directeur de recherche at the CNRS
• From 2008 to 2009: On leave
• From 2009 to 2018: Attached to the Institut de Mathématiques de Jussieu- Paris Rive Gauche (UMR 7586, CNRS, Universit e Paris Diderot, UPMC)

Prizes and honors

• Gold medal at the International Mathematical Olympiad, Canada, 1995
• Cours Peccot at the Collège de France, 2005
• Invited address at the ICMP, 2006
• Bronze medal of the CNRS, 2006
• Salem prize, 2006
• Wolff Memorial Lectures, Caltech, 2008
• Invited address at the ECM, 2008
• EMS prize, 2008
• Grand Prix Jacques Herbrand of the French Academy of Sciences, 2009
• Porter Lectures, Rice, 2010
• Plenary address at the ICM, 2010
• Blyth Lectures, Toronto, 2011
• Michael Brin prize, 2011
• IAMP Early Career Award, 2012
• Member, Academia Brasileira de Ciências, 2013
• Prêmio da SBM, 2013
• TWAS Prize, 2013
• Bellow Lectures, Northwestern, 2014
• Fields Medal, 2014
• Chevalier de la Légion d'honneur, 2015
• TWAS-Lenovo Science Prize, 2015
• Łojasiewicz Lecture, Krakow, 2017
• Grande Oficial da Ordem Nacional do Mérito Educativo, 2018
• Grã-Cruz da Ordem Nacional do Mérito Científico, 2018
• National Academy of Sciences, Foreign Member, United States, 2019

Service

Editorial boards:
• Ergodic Theory and Dynamical Systems (current)
• Mathematical Proceedings of the Cambridge Philosophical Society
• Publications Mathématiques de l'IHES
• Duke Mathematical Journal

• ERC Starting Grant "Quasiperiodic" (2010-2015)
• Swiss National Science Foundation, Funding scheme: Project funding (Div. I-III), Project "Dynamical Systems" (2020-2024)

Papers

Published

1. On the dynamics of the renormalization operator.
   Artur Avila, Marco Martens, Welington de Melo.
   In ''Global Analysis of Dynamical Systems: Festschrift dedicated to Floris Takens for his 60th birthday'', Ed. H.Broer, B. Krauskopf, G. Vegter, Institute of Physics Publishing, Bristol and Philadelphia, pp. 449-460, 2001.


2. Infinitesimal perturbations of rational maps.
   Artur Avila.
   Nonlinearity 15 (2002), 695-704.


3. A formula with applications to the theory of Lyapunov exponents.
   Artur Avila, Jairo Bochi.
   Israel Journal of Mathematics 131 (2002), 125-137.


4. Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative.
   Artur Avila, Carlos Gustavo Moreira.
   Astérisque 286 (2003), 81-118.


5. Regular or stochastic dynamics in real analytic families of unimodal maps.
   Artur Avila, Mikhail Lyubich, Welington de Melo.
   Inventiones Mathematicae 154 (2003), 451-550.


6. Robust transitivity and topological mixing for C¹-flows.
   Flavio Abdenur, Artur Avila, Jairo Bochi.
   Proceedings of the American Mathematical Society 132 (2004), 699-705.


7. Convergence of an exact quantization scheme.
   Artur Avila.
   Communications in Mathematical Physics 249 (2004), 305-318.


8. Quasisymmetric robustness of the Collet-Eckmann condition in the quadratic family.
   Artur Avila, Carlos Gustavo Moreira.
   Bulletin of the Brazilian Mathematical Society 35 (2004), 291-331.


9. Siegel disks with smooth boundaries.
   Artur Avila, Xavier Buff, Arnaud Chéritat.
   Acta Mathematica 193 (2004), 1-30.


10. Statistical properties of unimodal maps: the quadratic family.
    Artur Avila, Carlos Gustavo Moreira.
    Annals of Mathematics 161 (2005), 831-881.
    


11. Statistical properties of unimodal maps: physical measures, periodic points and pathological laminations.
    Artur Avila, Carlos Gustavo Moreira.
    Publications Mathématiques de l'IHÉS 101 (2005), 1-67.


12. Generic singular spectrum for ergodic Schrödinger operators.
    Artur Avila, David Damanik.
    Duke Mathematical Journal 130 (2005), 393-400.


13. Phase-Parameter relation and sharp statistical properties for general families of unimodal maps.
    Artur Avila, Carlos Gustavo Moreira.
    Contemporary Mathematics 389 (2005), 1-42.


14. Smoothness of solenoidal attractors.
    Artur Avila, Sébastien Gouëzel, Masato Tsujii.
    Discrete and Continuous Dynamical Systems 15 (2006), 21-35.


15. Examples of Feigenbaum Julia sets with small Hausdorff dimension.
    Arutr Avila, Mikhail Lyubich.
    In ''Dynamics on the Riemann Sphere. A Bodil Branner Festschrift'', Ed. Poul G. Hjorth, Carsten Lunde Petersen, European Mathematical Society (EMS), Zürich, pp. 71-87, 2006.


16. A generic C¹ map has no absolutely continuous invariant measure.
    Artur Avila, Jairo Bochi.
    Nonlinearity 19 (2006), 2717-2725.


17. Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles.
    Artur Avila, Raphaël Krikorian.
    Annals of Mathematics 164 (2006), 911-940.


18. Exponential mixing for the Teichmüller flow.
    Artur Avila, Sébastien Gouëzel, Jean-Christophe Yoccoz.
    Publications Mathématiques de l'IHÉS 104 (2006), 143-211.


19. Weak mixing for interval exchange transformations and translation flows.
    Artur Avila, Giovanni Forni.
    Annals of Mathematics 165 (2007), 637-664.


20. Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture.
    Artur Avila, Marcelo Viana.
    Acta Mathematica 198 (2007), 1-56.


21. Generic expanding maps without absolutely continuous invariant $\sigma$-finite measure.
    Artur Avila, Jairo Bochi.
    Mathematical Research Letters 14 (2007), 721-730.


22. Simplicity of Lyapunov spectra: a sufficient criterion.
    Artur Avila, Marcelo Viana.
    Portugaliae Mathematica 64 (2007), 311-376.


23. A uniform dichotomy for generic SL(2,R) cocycles over a minimal base.
    Artur Avila, Jairo Bochi.
    Bulletin de la Société Mathématique de France 135 (2007), 407-417.


24. Exponential decay of correlations for the Rauzy-Veech-Zorich induction map.
    Artur Avila, Alexander Bufetov.
    Fields Institute Communications 51 (2007), 203-211.


25. Hausdorff dimension and conformal measures of Feigenbaum Julia sets.
    Artur Avila, Mikhail Lyubich.
    Journal of the American Mathematical Society 21 (2008), 305-363.


26. Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling.
    Artur Avila, David Damanik.
    Inventiones Mathematicae 172 (2008), 439-453.


27. Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts.
    Artur Avila, Jairo Bochi, David Damanik.
    Duke Mathematical Journal 146 (2009), 253-280.


28. On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators.
    Artur Avila.
    Communications in Mathematical Physics 288 (2009), 907-918.


29. The ten Martini Problem.
    Artur Avila, Svetlana Jitomirskaya.
    Annals of Mathematics 170 (2009), 303-342.


30. Combinatorial rigidity for unicritical polynomials.
    Artur Avila, Jeremy Kahn, Mikhail Lyubich, Weixiao Shen.
    Annals of Mathematics 170 (2009), 783-797.


31. Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms.
    Artur Avila, Jairo Bochi, Amie Wilkinson.
    Annales Scientifiques de l'Ecole Normale Superieure 42 (2009), 931-979.


32. Uniform exponential growth for some SL(2,R) matrix products.
    Artur Avila, Thomas Roblin.
    Journal of Modern Dynamics 3 (2009), 549-554.


33. Density of positive Lyapunov exponents for quasiperiodic SL(2,R) cocycles in arbitrary dimension.
    Artur Avila.
    Journal of Modern Dynamics 3 (2009), 629-634.


34. Almost localization and almost reducibility.
    Artur Avila, Svetlana Jitomirskaya.
    Journal of the European Mathematical Society 12 (2010), 93-131.


35. Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectrum.
    Artur Avila, Barry Simon, Yoram Last.
    Analysis & PDE 3 (2010), 81-118.


36. Extremal Lyapunov exponents: an invariance principle and applications.
    Artur Avila, Marcelo Viana.
    Inventiones Mathematicae 181 (2010), 115-174.


37. Uniformly hyperbolic finite-valued SL(2,R)-cocycles.
    Artur Avila, Jairo Bochi, Jean-Christophe Yoccoz.
    Commentarii Mathematici Helvetici 85 (2010), 813-884.


38. On the regularization of conservative maps.
    Artur Avila.
    Acta Mathematica 205 (2010), 5-18.


39. Parapuzzle of the Multibrot set and typical dynamics of unimodal maps.
    Artur Avila, Mikhails Lyubich, Weixiao Shen.
    Journal of the European Mathematical Society 13 (2011), 27-56.


40. Hölder continuity of absolutely continuous spectral measures for one-frequency Schrödinger operators.
    Artur Avila, Svetlana Jitomirskaya.
    Communications in Mathematical Physics 301 (2011), 563-581.


41. Cohomological equations and invariant distributions for minimal circle diffeomorphisms.
    Artur Avila, Alejandro Kocsard.
    Duke Mathematical Journal 158 (2011), 501-536.


42. Density of positive Lyapunov exponents for SL(2,R) cocycles.
    Artur Avila.
    Journal of the American Mathematical Society 24 (2011), 999-1014.


43. A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies.
    Artur Avila, Bassam Fayad, Raphaël Krikorian.
    Geometric and Functional Analysis 21 (2011), 1001-1019.


44. Mixing for time-changes of Heisenberg nilflows.
    Artur Avila, Giovanni Forni, Corinna Ulcigrai.
    Journal of Differential Geometry 89 (2011), 369-410.


45. The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes.
    Artur Avila, Mikhail Lyubich.
    Publications Mathématiques de l'IHÉS 114 (2011), 171-223.


46. Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms.
    Artur Avila, Jairo Bochi.
    Transactions of the American Mathematical Society 364 (2012), 2883-2907.


47. Opening gaps in the spectrum of strictly ergodic Schrödinger operators.
    Artur Avila, Jairo Bochi, David Damanik.
    Journal of the European Mathematical Society 14 (2012), 61-106.


48. Exponential Mixing for the Teichmüller flow in the Space of Quadratic Differentials.
    Artur Avila, Maria João Resende.
    Commentarii Mathematici Helvetici 87 (2012), 589-638.


49. Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow.
    Artur Avila, Sébastien Gouëzel.
    Annals of Mathematics 178 (2013), 385-442.


50. SL(2,R)-invariant probability measures on the moduli spaces of translation surfaces are regular.
    Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz.
    Geometric and Functional Analysis 23 (2013), 1705-1729.


51. Holonomy invariance: rough regularity and applications to Lyapunov exponents.
    Artur Avila, Jimmy Santamaria, Marcelo Viana.
    Astérisque 358 (2013), 13-74.


52. On rigidity of critical circle maps.
    Artur Avila.
    Bulletin of the Brazilian Mathematical Society 44 (2013), 601-609.


53. Singular density of states measure for subshift and quasi-periodic Schrödinger operators.
    Artur Avila, David Damanik, Zhenghe Zhang.
    Communications in Mathematical Physics 330 (2014), 469-498.


54. Complex one-frequency cocycles.
    Artur Avila, Svetlana Jitomirskaya, Christian Sadel.
    Journal of the European Mathematical Society 16 (2014), 1915-1935.


55. On manifolds supporting distributionally uniquely ergodic diffeomorphisms.
    Artur Avila, Bassam Fayad, Alejandro Kocsard.
    Journal of Differential Geometry 99 (2015), 191-213.


56. On the Kotani-Last and Schrödinger conjectures.
    Journal of the American Mathematical Society 28 (2015), 579-616.


57. Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows.
    Artur Avila, Marcelo Viana, Amie Wilkinson.
    Journal of the European Mathematical Society 17 (2015), 1435-1462.


58. The visits to zero of a random walk driven by an irrational rotation.
    Artur Avila, Dmitry Dolgopyat, Eduard Duryev, Omri Sarig.
    Israel Journal of Mathematics 207 (2015), 653-717.


59. Monotonic cocycles.
    Artur Avila, Raphaël Krikorian.
    Inventiones Mathematicae 202 (2015), 271-331.


60. Global theory of one-frequency Schrödinger operators.
    Artur Avila.
    Acta Mathematica 215 (2015) 1-54.
    This version combines the preprints
    Global theory of one-frequency Schrödinger operators I: stratifed analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity,
    Global theory of one-frequency Schrödinger operators II: acriticality and finiteness of phase transitions for typical potentials.


61. Weak mixing directions in non-arithmetic Veech surfaces.
    Artur Avila, Vincent Delecroix.
    Journal of the American Mathematical Society 29 (2016), 1167-1208.


62. Towers for commuting endomorphisms, and combinatorial applications.
    Artur Avila, Pablo Candela.
    Annales de l'Institut Fourier 66 (2016), 1529-1544.


63. An integrable deformation of an ellipse of small eccentricity is an ellipse.
    Artur Avila, Jacopo de Simoi, Vadim Kaloshin.
    Annals of Mathematics 184 (2016), 527-558.
    


64. On the Hausdorff dimension of the Rauzy gasket.
    Artur Avila, Pascal Hubert, Alexandra Skripchenko.
    Bulletin de la Société Mathématique de France 144 (2016), 539-568.


65. Diffusion for chaotic plane sections of 3-periodic surfaces.
    Artur Avila, Pascal Hubert, Alexandra Skripchenko.
    Inventiones Mathematicae 206 (2016), 109-146.
    


66. Diffeomorphisms with positive metric entropy.
    Artur Avila, Sylvain Crovisier, Amie Wilkinson.
    Publications Mathématiques de l'IHÉS 124 (2016), 319-347.
    


67. Spectral Theory of Extended Harper's Model and a Question by Erdős and Szekeres.
    Artur Avila, Svetlana Jitomirskaya, Chris Marx.
    Inventiones Mathematicae 210 (2017), 283-339.
    


68. Sharp Phase transitions for the almost Mathieu operator.
    Artur Avila, Jiangong You, Qi Zhou.
    Duke Mathematical Journal 166 (2017), 2697-2718.


69. Symplectic and isometric SL(2,R)-invariant subbundles of the Hodge bundle.
    Artur Avila, Alex Eskin, Martin Möller.
    Journal für die reine und angewandte Mathematik 732 (2017), 1-20.


70. Second phase transition line.
    Artur Avila, Svetlana Jitomirskaya, Qi Zhou.
    Mathematische Annalen 370 (2018), 271-285.


71. Zorich conjecture for hyperelliptic Rauzy-Veech groups.
    Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz.
    Mathematische Annalen 370 (2018), 785-809.


72. Livsic theorem for diffeomorphism cocycles.
    Artur Avila, Alejandro Kocsard, Xiao-Chuan Liu.
    Geometric and Functional Analysis 28 (2018), 943-964.


73. Statistical properties of quadratic polynomials with a neutral fixed point.
    Artur Avila, Davoud Cheraghi.
    Journal of the European Mathematical Society 20 (2018), 2005-2062.


74. Weak mixing properties of interval exchange transformations and translation flows.
    Artur Avila, Martin Leguil.
    Bulletin de la Société Mathématique de France 146 (2018) 391-426.


75. The Kontsevich-Zorich cocycle over Veech-McMullen family of symmetric translation surfaces.
    Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz.
    Journal of Modern Dynamics 14 (2019), 21-54.


76. Some monoids of Pisot matrices.
    Artur Avila, Vincent Delecroix.
    In ''New trends in one-dimensional dynamics'', Ed. Maria José Pacifico, Pablo Guarino, Springer Proceedings in Mathematics & Statistics, pp. 21-30, 2019.


77. Recurrence for the wind-tree model.
    Artur Avila, Pascal Hubert.
    Annales de l'Institut Henri Poincaré - Analyse non linéaire 37 (2020), 1-11.


78. Smooth Siegel disks everywhere.
    Artur Avila, Xavier Buff, Arnaud Chéritat.
    Astérisque 416 (2020), 133-180.


79. Stable accessibility with 2-dimensional center.
    Artur Avila, Marcelo Viana.
    Astérisque 416 (2020), 301-320.


80. On mixing diffeomorphisms of the disk.
    Artur Avila, Bassam Fayad, Patrice Le Calvez, Disheng Xu, Zhiyuan Zhang.
    Inventiones Mathematicae 220 (2020), 673-714


81. Invariant graphs and spectral type of Schrödinger operators.
    Artur Avila, Konstantin Khanin, Martin Leguil.
    Pure and Applied Functional Analysis 5 (2020), 1257-1277.


82. C¹-density of stable ergodicity.
    Artur Avila, Sylvain Crovisier, Amie Wilkinson.
    Advances in Mathematics 379 (2021), 68pp.


83. Mixing for smooth time-changes of general nilflows.
    Artur Avila, Giovanni Forni, Davide Ravotti, Corinna Ulcigrai.
    Advances in Mathematics 385 (2021), 65pp.


84. Lebesgue measure of Feigenbaum Julia sets.
    Artur Avila, Mikhail Lyubich.
    Annals of Mathematics 195 (2022), 1-88.


85. Absolute continuity, Lyapunov exponents, and rigidity II : systems with compact center leaves.
    Artur Avila, Marcelo Viana, Amie Wilkinson.
    Ergodic Theory & Dynamical Systems 42 (2022), 437-490.


86. Non-dirrerentiable irrational curves for C¹ twist map.
    Artur Avila, Bassam Fayad.
    Ergodic Theory & Dynamical Systems 42 (2022), 491-499.

Accepted for publication

87. Analytic maps of parabolic and elliptic type with trivial centralisers.
    Artur Avila, Davoud Cheraghi, Alexander Eliad.
    To appear in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire.


88. Symplectomorphisms with positive metric entropy.
    Artur Avila, Sylvain Crovisier, Amie Wilkinson.
    To appear in: Proceedings of the London Mathematical Society.

Submitted for publication

89. Schrödinger operators with potentials generated by hyperbolic transformations: I. Positivity of the Lyapunov exponent.
    Artur Avila, David Damanik, Zhenghe Zhang.


90. Quantitative weak mixing for interval exchange transformations.
    Artur Avila, Giovanni Forni, Pedram Safaee.


91. On the abominable properties of the almost Mathieu operator with well approximated frequencies.
    Artur Avila, Yoram Last, Mira Shamis, Qi Zhou.

Preprints

92. Smooth Siegel disks via semicontinuity: a remark on a proof of Buff and Cheritat.
    Artur Avila.


93. The absolutely continuous spectrum of the almost Mathieu operator.
    Artur Avila.


94. On point spectrum with critical coupling.
    Artur Avila.


95. Distortion elements in $Diff^\infty(R/Z)$.
    Artur Avila.


96. Almost reducibility and absolute continuity I.
    Artur Avila.

Expository

97. Bifurcations d'applications unimodales.
    Artur Avila.
    C. R. Acad. Sci. Paris, Ser. I 334 (2002), 483-488.


98. Bifurcations of unimodal maps.
    Artur Avila, Carlos Gustavo Moreira.
    In Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Publicazioni della Classe di Scienze, Scuola Normale Superieure, Pisa. Centro di Ricerca Matematica ''Ennio De Giorgi'': Proceedings, 2003, pp. 1-22.


99. Dynamics in the moduli space of Abelian di erentials.
    Artur Avila, Marcelo Viana.
    Portugaliae Mathematica 62 (2005), 531-547.


100. Solving the ten martini problem.
     Artur Avila, Svetlana Jitomirskaya.
     Lecture Notes in Physics 690 (2006), 5-16.


101. Chaoticity of the Teichmüller flow.
     Artur Avila.
     Clay Mathematics Proceedings 10 (2010), 321-338.


102. Dynamics of renormalization operators.
     Artur Avila.
     In ''Proceedings of the International Congress of Mathematicians 2010. Volume 1.'' World Scientific Publishing, pp. 154-175.


103. Cocycles over partially hyperbolic maps.
     Artur Avila, Marcelo Viana, Amie Wilkinson.
     Astérisque 358 (2013), 1-12.

Anlässe

Zurich Dynamics Conference
with Manfred Einsiedler and Çağri Sert
June 12-16, 2023, in Zurich

Zurich Dynamics School
with Manfred Einsiedler and Çağri Sert
June 8-10, 2023, in Zurich