AVILA, ARTUR / Matemàticas / Investigadores

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Chargé de Recherche, CNRS U.M.R. 7599, "Probabilités et Modèles Aléatoires", Laboratoire de Probabilités, Université Paris VI, Paris, France.

Artur Avila received his Ph.D. in 2001 under the direction of Welington de Melo at the Instituto Nacional de Matemática Pura e Aplicada (IMPA), in Rio de Janeiro, Brazil. His thesis, "Bifurcations of unimodal maps: the topological and metric picture," generalized the regular or stochastic dichotomy from the quadratic family to any non-trivial family of real analytic unimodal maps. Since then he has made numerous outstanding contributions to one-dimensional and holomorphic dynamics, spectral theory of the Schroedinger operator, and ergodic theory of interval exchange transformations and the associated Teichmüller flow.

Gold medal at the International Mathematical Olympiad. Canada, 1995.
Cours Peccot at the Collège de France, 2005.
Bronze medal of the CNRS, 2006.
Salem prize, 2006.
Wolff Memorial Lectures at Caltech, 2008.
EMS prize, 2008.

On the dynamics of the renormalization operator.
A. Avila, Marco Martens and 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.

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).

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

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

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

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

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.

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

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

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.

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

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

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.

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

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

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

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

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.

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

A generic $C^1$ map has no absolutely continuous invariant measure.
A. Avila, J. Bochi.
Nonlinearity 19 (2006), 2717-2725.

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

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

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

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

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

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

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.

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

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

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.

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