# Artur Avila

#8,801

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Brazilian and French mathematician

## Artur Avila's AcademicInfluence.com Rankings

Artur Avilamathematics Degrees

Mathematics

#987

World Rank

#1703

Historical Rank

Measure Theory

#3330

World Rank

#3934

Historical Rank

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Mathematics

## Why Is Artur Avila Influential?

(Suggest an Edit or Addition)According to Wikipedia, Artur Avila Cordeiro de Melo is a Brazilian and naturalized French mathematician working primarily in the fields of dynamical systems and spectral theory. He is one of the winners of the 2014 Fields Medal, being the first Latin American and lusophone to win such an award. He has been a researcher at both the IMPA and the CNRS . He has been a professor at the University of Zurich since September 2018.

## Artur Avila's Published Works

### Published Works

- The Ten Martini Problem (2009) (296)
- Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodinger cocycles (2003) (197)
- Exponential mixing for the Teichmüller flow (2005) (187)
- Weak mixing for interval exchange transformations and translation flows (2004) (183)
- Almost localization and almost reducibility (2008) (170)
- Global theory of one-frequency Schrödinger operators (2015) (164)
- Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture (2005) (163)
- Extremal Lyapunov exponents: an invariance principle and applications (2010) (134)
- Regular or stochastic dynamics in real analytic families of unimodal maps (2003) (108)
- Sharp phase transitions for the almost Mathieu operator (2015) (96)
- A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies (2010) (85)
- Almost reducibility and absolute continuity I (2010) (75)
- An integrable deformation of an ellipse of small eccentricity is an ellipse (2014) (68)
- The absolutely continuous spectrum of the almost Mathieu operator (2008) (65)
- Spectral theory of extended Harper’s model and a question by Erdős and Szekeres (2016) (64)
- Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows (2011) (64)
- Statistical properties of unimodal maps: the quadratic family (2000) (62)
- Symplectic and Isometric SL(2,R) invariant subbundles of the Hodge bundle (2012) (61)
- On the regularization of conservative maps (2008) (61)
- Complex one-frequency cocycles (2013) (60)
- A Kam Scheme for SL(2, $${{\mathbb R}}$$) Cocycles with Liouvillean Frequencies (2010) (57)
- Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling (2007) (57)
- On the Spectrum and Lyapunov Exponent of Limit Periodic Schrödinger Operators (2008) (54)
- Hausdorff dimension and conformal measures of Feigenbaum Julia sets (2004) (53)
- Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow (2010) (51)
- Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles (2008) (48)
- HOLONOMY INVARIANCE: ROUGH REGULARITY AND APPLICATIONS TO LYAPUNOV EXPONENTS par (2012) (48)
- A formula with some applications to the theory of Lyapunov exponents (2001) (48)
- Siegel disks with smooth boundaries (2004) (47)
- Combinatorial rigidity for unicritical polynomials (2005) (47)
- Diffeomorphisms with positive metric entropy (2014) (45)
- Simplicity of Lyapunov spectra: a sufficient criterion (2006) (45)
- Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts (2007) (42)
- Generic Singular Spectrum For Ergodic Schrödinger Operators (2005) (42)
- Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity (2009) (42)
- Density of positive Lyapunov exponents for SL(2,R) cocycles (2010) (41)
- Monotonic cocycles (2013) (40)
- On the Hausdorff dimension of the Rauzy gasket (2013) (40)
- The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes (2010) (39)
- Opening gaps in the spectrum of strictly ergodic (2009) (38)
- On rigidity of critical circle maps (2013) (38)
- Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum (2008) (37)
- Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative (2001) (36)
- Diffusion for chaotic plane sections of 3-periodic surfaces (2014) (36)
- C1 density of stable ergodicity (2017) (36)
- Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with absolutely continuous spectrum (2010) (36)
- Nonuniform Hyperbolicity, Global Dominated Splittings and Generic Properties of Volume-Preserving Diffeomorphisms (2009) (34)
- Smoothness of solenoidal attractors (2005) (33)
- A generic C1 map has no absolutely continuous invariant probability measure (2006) (33)
- Exponential Mixing for the Teichmuller flow in the Space of Quadratic Differentials (2009) (31)
- Hölder Continuity of Absolutely Continuous Spectral Measures for One-Frequency Schrödinger Operators (2009) (30)
- Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators (2013) (27)
- Solving the Ten Martini Problem (2005) (26)
- Mixing for the time-changes of Heisenberg nilflows (2010) (26)
- The visits to zero of a random walk driven by an irrational rotation (2015) (25)
- Weak mixing directions in non-arithmetic Veech surfaces (2013) (25)
- On the Kotani-Last and Schrodinger conjectures (2012) (25)
- Recurrence for the wind-tree model (2017) (23)
- Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms (2008) (23)
- Cocycles over Partially Hyperbolic Maps (2013) (23)
- GLOBAL THEORY OF ONE-FREQUENCY SCHRÖDINGER OPERATORS II: ACRITICALITY AND FINITENESS OF PHASE TRANSITIONS FOR TYPICAL POTENTIALS (2013) (22)
- Statistical properties of unimodal maps: physical measures, periodic orbits and pathological laminations (2003) (21)
- Livšic theorem for diffeomorphism cocycles (2017) (20)
- PARAPUZZLE OF THE MULTIBROT SET AND TYPICAL DYNAMICS OF UNIMODAL MAPS (2008) (20)
- Cohomological equations and invariant distributions for minimal circle diffeomorphisms (2010) (20)
- Phase-parameter relation and sharp statistical properties for general families of unimodal maps (2003) (17)
- A uniform dichotomy for generic $SL(2,R)$ cocycles over a minimal base (2006) (16)
- Lebesgue measure of Feigenbaum Julia sets (2015) (16)
- Infinitesimal perturbations of rational maps (2001) (15)
- Statistical Properties of Unimodal Maps (2000) (14)
- Second phase transition line (2016) (14)
- Mixing for smooth time-changes of general nilflows (2019) (13)
- Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves (2021) (13)
- A uniform dichotomy for generic ${\rm SL}(2,{\mathbb {R}})$ cocycles over a minimal base (2007) (13)
- Convergence of an Exact Quantization Scheme (2003) (12)
- Stable accessibility with 2-dimensional center (2020) (12)
- On the dynamics of the renormalization operator (2001) (11)
- Exponential decay of correlations for the Rauzy-Veech-Zorich induction map (2006) (11)
- Zorich conjecture for hyperelliptic Rauzy–Veech groups (2016) (11)
- Dynamics in the moduli space of Abelian differentials (2005) (11)
- Robust transitivity and topological mixing for C^1-flows (2002) (11)
- $${SL(2, \mathbb{R})}$$SL(2,R) -invariant probability measures on the moduli spaces of translation surfaces are regular (2013) (10)
- Some Monoids of Pisot Matrices (2015) (10)
- Density of positive Lyapunov exponents for quasiperiodic SL(2, R) -cocycles in arbitrary dimension (2010) (9)
- On mixing diffeomorphisms of the disk (2015) (9)
- Examples of Feigenbaum Julia sets with small Hausdorff dimension (2004) (8)
- On the abominable properties of the almost Mathieu operator with well approximated frequencies (2021) (8)
- Distortion elements in $Diff^\infty(R/Z)$ (2008) (8)
- Generic singular continuous spectrum for ergodic Schr\"odinger operators (2004) (7)
- Statistical properties of quadratic polynomials with a neutral fixed point (2012) (7)
- Towers for commuting endomorphisms, and combinatorial applications (2015) (7)
- Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent (2020) (7)
- On manifolds supporting distributionally uniquely ergodic diffeomorphisms (2012) (7)
- On mixing diffeomorphisms of the disc (2019) (7)
- Weak mixing properties of interval exchange transformations and translation flows (2016) (6)
- Bifurcations of unimodal maps (2003) (5)
- Uniform exponential growth for some SL(2, R) matrix products (2010) (5)
- Smooth Siegel disks via semicontinuity: a remark on a proof of Buff and Cheritat (2003) (5)
- Dynamics of Renormalization Operators (2011) (5)
- The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces (2017) (4)
- Generic expanding maps without absolutely continuous invariant $\sigma$-finite measure (2006) (4)
- Quasisymmetric robustness of the Collet-Eckmann condition in the quadratic family (2001) (4)
- Symplectomorphisms with positive metric entropy (2019) (3)
- Quantitative weak mixing for interval exchange transformations (2021) (3)
- Bifurcations d'applications unimodales (2002) (2)
- Invariant graphs and spectral type of Schr\"odinger operators (2020) (2)
- Spectral theory of extended Harper’s model and a question by Erdős and Szekeres (2017) (2)
- MECHANISMS FOR CHAOS (2019) (1)
- Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators (2014) (1)
- Non-differentiable irrational curves for $C^1$ twist map (2019) (1)
- ON LYAPUNOV EXPONENTS PART I (2008) (1)
- Analytic maps of parabolic and elliptic type with trivial centralisers (2020) (1)
- QUASIPERIODIC SL(2, R) COCYCLES WHICH ARE NOT HOMOTOPIC TO THE IDENTITY (2006) (1)
- Diffusion for chaotic plane sections of 3-periodic surfaces (2016) (0)
- Second phase transition line (2017) (0)
- Awards at the International Congress of Mathematicians at Seoul (Korea) 2014 News release from the International Mathematical Union (IMU) (2015) (0)
- COCYCLES OVER PARTIALLY HYPERBOLIC MAPS par (2012) (0)
- ON THE SUBADDITIVE ERGODIC THEOREM (2009) (0)
- Smooth Siegel disks everywhere (2019) (0)
- On rigidity of critical circle maps (2013) (0)
- Zorich conjecture for hyperelliptic Rauzy–Veech groups (2017) (0)
- Livšic theorem for diffeomorphism cocycles (2018) (0)
- Ju l 2 00 6 SIMPLICITY OF LYAPUNOV SPECTRA : A SUFFICIENT CRITERION (2018) (0)
- COCYCLES OVER A MINIMAL BASE (2008) (0)
- Monotonic cocycles (2015) (0)
- The Spectrum of Schrödinger Operators with Randomly Perturbed Ergodic Potentials (2022) (0)
- Série SIMPLICITY OF LYAPUNOV SPECTRA : A SUFFICIENT CRITERION (2007) (0)

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## Other Resources About Artur Avila

## What Schools Are Affiliated With Artur Avila?

Artur Avila is affiliated with the following schools: