# Prof. Dr. Artur Avila

Institut für Mathematik
Universität Zürich
Winterthurerstrasse 190
CH-8057 Zürich

E-Mail:
Phone: +41 44 635 58 80
Fax: +41 44 635 57 06
Office: Y27K18
Field of Research:
Dynamical Systems
Working group:
Secretary:
E-Mail:
Phone:
Myriam Frank-Hilpert

+41 44 635 58 32

## Vita

### 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
• Lojasiewicz Lecture, Krakow, 2017
• Election as Foreign Associate of the U.S. National Academy of Sciences, 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 (2010-2015)

## Papers

### Published

1. On the dynamics of the renormalization operator.
A. 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.
Nonlinearity 15 (2002), 695-704.

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

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

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

6. Robust transitivity and topological mixing for $C^1$-flows.
Flavio Abdenur, A. Avila, J. Bochi.
Proceedings of the American Mathematical Society 132 (2004), 699-705.

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

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

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

10. Statistical properties of unimodal maps: the quadratic family.
A. Avila, C. G. Moreira.
Annals of Mathematics 161 (2005), 831-881.

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

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

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

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

15. Examples of Feigenbaum Julia sets with small Hausdorff dimension.
A. Avila, M. 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 $C^1$ map has no absolutely continuous invariant measure.
A. Avila, J. Bochi.
Nonlinearity 19 (2006), 2717-2725.

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

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

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

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

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

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

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

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

25. Hausdorff dimension and conformal measures of Feigenbaum Julia sets.
A. Avila, M. 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.
A. Avila, D. Damanik.
Inventiones Mathematicae 172 (2008), 439-453.

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

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

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

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

31. Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms.
A. Avila, J. 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.
A. 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.
Journal of Modern Dynamics 3 (2009), 629-634.

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

35. Extremal Lyapunov exponents: an invariance principle and applications.
A. Avila, M. Viana.
Inventiones Mathematicae 181 (2010), 115-174.

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

37. Uniformly hyperbolic finite-valued SL(2,R)-cocycles.
A. Avila, J. Bochi, J.-C. Yoccoz.
Commentarii Mathematici Helvetici 85 (2010), 813-884.

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

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

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

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

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

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

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

45. Mixing for time-changes of Heisenberg nilflows.
A. Avila, G. Forni, Corinna Ulcigrai.
Journal of Differential Geometry 89 (2011), 369-410.

46. Opening gaps in the spectrum of strictly ergodic Schrödinger operators.
A. Avila, J. Bochi, D. Damanik.
Journal of the European Mathematical Society 14 (2012), 61-106.

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

48. Exponential Mixing for the Teichmüller flow in the Space of Quadratic Differentials.
A. 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.
A. Avila, S. Gouëzel.
Annals of Mathematics 178 (2013), 385-442.

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

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

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

53. Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators.
A. Avila, D. Damanik, Zhenghe Zhang.
Communications in Mathematical Physics 330 (2014), 469-498.

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

55. On manifolds supporting distributionally uniquely ergodic diffeomorphisms.
A. Avila, B. Fayad, A. 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.
A. Avila, M. Viana, A. 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.
A. Avila, Dmitry Dolgopyat, Eduard Duryev, Omri Sarig.
Israel Journal of Mathematics 207 (2015), 653-717.

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

60. Global theory of one-frequency Schrödinger operators.
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.
A. Avila, Vincent Delecroix.
Journal of the American Mathematical Society 29 (2016), 1167-1208.

62. Towers for commuting endomorphisms, and combinatorial applications.
A. 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.
A. Avila, Jacopo de Simoi, Vadim Kaloshin.
Annals of Mathematics 184 (2016), 527-558.

64. On the Hausdorff dimension of the Rauzy gasket.
A. 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.
A. Avila, P. Hubert, A. Skripchenko.
Inventiones Mathematicae 206 (2016), 109-146.

66. Diffeomorphisms with positive metric entropy.
A. Avila, Sylvain Crovisier, A. 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.
A. Avila, S. Jitomirskaya, Chris Marx.
Inventiones Mathematicae 210 (2017), 283-339.

68. Sharp Phase transitions for the almost Mathieu operator.
A. 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.
A. Avila, Alex Eskin, Martin Möller.
Journal für die reine und angewandte Mathematik 732 (2017), 1-20.

70. Second phase transition line.
A. Avila, S. Jitomirskaya, Q. Zhou.
Mathematische Annalen 370 (2018), 271-285.

71. Zorich conjecture for hyperelliptic Rauzy-Veech groups.
A. Avila, C. Matheus, J.-C. Yoccoz.
Mathematische Annalen 370 (2018), 785-809.

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

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

75. Recurrence for the wind-tree model.
A. Avila, P. Hubert.
To appear in Annales de l'Institut Henri Poincaré - Analyse non linéaire.

76. Weak mixing properties of interval exchange transformations and translation flows.
A. Avila, Martin Leguil.
To appear in Bulletin de la Société Mathématique de France.
77. ### Submitted for publication

78. On mixing diffeomorphisms of the disk.
A. Avila, B. Fayad, Patrice Le Calvez, Disheng Xu, Zhiyuan Zhang.

79. Lebesgue measure of Feigenbaum Julia sets.
A. Avila, M. Lyubich.

80. The Kontsevich-Zorich cocycle over Veech-McMullen family of symmetric translation surfaces.
A. Avila, C. Matheus, J.-C. Yoccoz.

81. Smooth Siegel disks everywhere.
A. Avila, X. Buff, A. Chéritat.

82. Some monoids of Pisot matrices.
A. Avila, V. Delecroix.

83. $C^1$-density of stable ergocicity.
A. Avila, S. Crovisier, A. Wilkinson.

84. Stable accessibility with 2-dimensional center.
A. Avila, M. Viana.
85. ### Preprints

86. Smooth Siegel disks via semicontinuity: a remark on a proof of Buff and Cheritat.
Not intended for publication.

87. The absolutely continuous spectrum of the almost Mathieu operator.

88. On point spectrum with critical coupling.
Not intended for publication.

89. Distortion elements in $Diff^\infty(R/Z)$.

90. Almost reducibility and absolute continuity I.
91. ### In preparation

92. Hausdorff dimension and the quadratic family.
A. Avila, C. G. Moreira.
93. ### Expository

94. Bifurcations d'applications unimodales.
C. R. Acad. Sci. Paris, Ser. I 334 (2002), 483-488. (French/English).
There is a corrigendum (which only affects the french version).

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

96. Dynamics in the moduli space of Abelian differentials.
A. Avila, Marcelo Viana.
Portugaliae Mathematica 62 (2005), 531-547.

97. Solving the Ten Martini Problem.
A. Avila, Svetlana Jitomirskaya.
Lecture Notes in Physics 690 (2006), 5-16.

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

99. Dynamics of renormalization operators.
In Proceeding of the International Congress of Mathematicians 2010. Volume 1.'' World Scientific Publishing, pp. 154-175.

100. Cocycles over partially hyperbolic maps.
A. Avila, M. Viana, A. Wilkinson.
Astérisque 358 (2013), 1-12.

## Seminars & Lectures

Research seminars
Ergodic Theory and Dynamical Systems Seminar
Mo
13.45-14.45
Menny Akka, Artur Avila, Manfred Einsiedler, Alexander Gorodnik, Cagri Sert, Corinna Ulcigrai

Lectures
Renormalization dynamics and ergodic theory of interval exchange transformations
Di
13.00-14.45

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