Exponential Mixing for the Teichmuller flow in the Space of Quadratic Differentials (2009)
Avila, Artur, Resende, Maria Joao
We consider the Teichmuller flow on the unit cotangent bundle of the moduli space of compact Riemann surfaces with punctures. We show that it is exponentially mixing for the Ratner class of...
We study Schrodinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials....
Opening Gaps in the Spectrum of Strictly Ergodic Schr\"odinger Operators (2009)
Avila, Artur, Bochi, Jairo, Damanik, David
We consider Schr\"odinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the...
Uniform exponential growth for some SL(2,R) matrix products (2009)
Given a hyperbolic matrix $H\in SL(2,\R)$, we prove that for almost every $R\in SL(2,\R)$, any product of length $n$ of $H$ and $R$ grows exponentially fast with $n$ provided the matrix $R$ occurs...
Uniform exponential growth for some SL(2,R) matrix products (2009)
Given a hyperbolic matrix $H\in SL(2,\R)$, we prove that for almost every $R\in SL(2,\R)$, any product of length $n$ of $H$ and $R$ grows exponentially fast with $n$ provided the matrix $R$ occurs...
Avila, Artur, Bochi, Jairo, Wilkinson, Amie
We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns--Wilkinson and Avila--Santamaria--Viana. Combining this new...
Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum (2008)
Avila, Artur, Last, Yoram, Simon, Barry
By combining some ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for OPRL in the a.c. spectral region is implied by convergence of $\frac{1}{n}...
The absolutely continuous spectrum of the almost Mathieu operator (2008)
We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon's list of Schr\"odinger operator...
On the regularization of conservative maps (2008)
We show that smooth maps are $C^1$-dense among $C^1$ volume preserving maps.
Distortion elements in $Diff^\infty(R/Z)$ (2008)
We consider the group of smooth diffeomorphisms of the circle. We show that any recurrent $f$ (in the sense that $\{f^n\}_{n \in Z}$ is not discrete) is in fact a distortion element (in the sense...
Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles (2008)
Avila, Artur, Bochi, Jairo, Yoccoz, Jean-Christophe
We consider finite families of SL(2,R) matrices whose products display uniform exponential growth. These form open subsets of (SL(2,R))^N, and we study their components, boundary, and complement. We...
On the spectrum and Lyapunov exponent of limit periodic Schrodinger operators (2008)
We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schr\"odinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero...
Almost localization and almost reducibility (2008)
Avila, Artur, Jitomirskaya, Svetlana
We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schr\"odinger cocycles associated to a non-perturbatively small analytic potential...
Parapuzzle of the Multibrot set and typical dynamics of unimodal maps (2008)
Avila, Artur, Lyubich, Mikhail, Shen, Weixiao
We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely...
We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum...
Avila, Artur, Bochi, Jairo, Damanik, David
We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle...
Weak mixing for interval exchange transformations and translation flows (2007)
We prove that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation ow on a typical translation surface of...
A uniform dichotomy for generic $SL(2,R)$ cocycles over a minimal base (2006)
We consider continuous $SL(2,R)$-cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform...
Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles (2006)
Avila, Artur, Krikorian, Raphaël
We show that for almost every frequency $\alpha \in \R \setminus \Q$, for every $C^\omega$ potential $v:\R/\Z \to \R$, and for almost every energy $E$ the corresponding quasiperiodic Schrödinger...
Exponential decay of correlations for the Rauzy-Veech-Zorich induction map (2006)
Avila, Artur, Bufetov, Alexander
We prove exponential mixing for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations (Theorem 3)
Generic expanding maps without absolutely continuous invariant $\sigma$-finite measure (2006)
We show that a $C^1$-generic expanding map of the circle has no absolutely continuous invariant $\sigma$-finite measure.
Simplicity of Lyapunov spectra: a sufficient criterion (2006)
We exhibit an explicit sufficient condition for the Lyapunov exponents of a linear cocycle over a Markov map to have multiplicity 1. This builds on work of Guivarc'h-Raugi and Gol'dsheid-Margulis,...
A generic $C^1$ map has no absolutely continuous invariant probability measure (2006)
Let $M$ be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension $d \ge 1$. We consider the set of $C^1$ maps $f:M\to M$ which have no absolutely continuous (with...
Exponential mixing for the Teichmuller flow (2005)
Avila, Artur, Gouezel, Sebastien, Yoccoz, Jean-Christophe
We study the dynamics of the Teichmuller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the...
Generic Singular Spectrum For Ergodic Schrödinger Operators (2005)
We consider Schrödinger operators with ergodic potential $V_\omega(n)=f(T^{n}(\omega))$, $n \in \bb{Z}$, $\omega \in \Omega$, where $T:\Omega \to \Omega$ is a nonperiodic homeomorphism. We show that...
Smoothness of solenoidal attractors (2005)
Avila, Artur, Gouezel, Sebastien, Tsujii, Masato
We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$: $ T:S^{1}\times \R\to S^1\times \R, T(x,y)=(\ell x,...
Statistical properties of unimodal maps: the quadratic family (2005)
Avila, Artur, Moreira, Carlos Gustavo
We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps...
Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture (2005)
We prove the Zorich-Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichm\"uller flow on (any connected component of a stratum of) the moduli space of Abelian differentials on...
Combinatorial rigidity for unicritical polynomials (2005)
Avila, Artur, Kahn, Jeremy, Lyubich, Mikhail, Shen, Weixiao
We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the...
The Ten Martini Problem (2005)
Avila, Artur, Jitomirskaya, Svetlana
We prove the conjecture (known as the ``Ten Martini Problem'' after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all non-zero values of the coupling and all...
Smoothness of solenoidal attractors (2005)
Avila, Artur, Gouezel, Sebastien, Tsujii, Masato
We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$: $ T:S^{1}\times \R\to S^1\times \R, T(x,y)=(\ell x,...
Statisical properties of unimodal maps: the quadratic family (2005)
Avila, Artur, Moreira, Carlos Gustavo
We prove that almost every nonregular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps...
Smoothness of solenoidal attractors (2005)
Avila, Artur, Gouezel, Sebastien, Tsujii, Masato
We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$: $ T:S^{1}\times \R\to S^1\times \R, T(x,y)=(\ell x,...
Smoothness of solenoidal attractors (2005)
Avila, Artur, Gouezel, Sebastien, Tsujii, Masato
We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$: $ T:S^{1}\times \R\to S^1\times \R, T(x,y)=(\ell x,...
Abstract. We prove the Zorich-Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichmüller flow on (any connected component of a stratum of) the moduli space of Abelian...
Generic singular continuous spectrum for ergodic Schr\"odinger operators (2004)
We consider Schr\"odinger operators with ergodic potential $V_\omega(n)=f(T^n(\omega))$, $n \in \Z$, $\omega \in \Omega$, where $T:\Omega \to \Omega$ is a non-periodic homeomorphism. We show that for...
Hausdorff dimension and conformal measures of Feigenbaum Julia sets (2004)
Avila, Artur, Lyubich, Mikhail
We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the ``hairiness phenomenon'', there exist many Feigenbaum Julia sets $J(f)$ whose...
Examples of Feigenbaum Julia sets with small Hausdorff dimension (2004)
Avila, Artur, Lyubich, Mikhail
We give examples of infinitely renormalizable quadratic polynomials $F_c: z\maps to z^2+c$ with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrar y close to 1. The...
Weak mixing for interval exchange transformations and translation flows (2004)
We show that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a surface of genus $g \geq 2$...
Reducibility or non-uniform hyperbolicity for quasiperiodic Schrodinger cocycles (2003)
Avila, Artur, Krikorian, Raphael
We show that for almost every frequency alpha \in \R \setminus \Q, for every C^omega potential v:\R/\Z \to R, and for almost every energy E the corresponding quasiperiodic Schrodinger cocycle is...
Convergence of an exact quantization scheme (2003)
It has been shown by Voros \cite {V} that the spectrum of the one-dimensional homogeneous anharmonic oscillator (Schr\"odinger operator with potential $q^{2M}$, $M>1$) is a fixed point of an explicit...
Avila, Artur, Moreira, Carlos Gustavo
In this work, we relate the geometry of chaotic attractors of typical analytic unimodal maps to the behavior of the critical orbit. Our main result is an explicit formula relating the combinatorics...
Avila, Artur, Moreira, Carlos Gustavo
We obtain estimates relating the phase space and the parameter space of analytic families of unimodal maps. Using those estimates, we show that typical analytic unimodal maps admit a quasiquadratic...
Bifurcations of unimodal maps (2003)
Avila, Artur, Moreira, Carlos Gustavo
We review recent results that lead to a very precise understanding of the dynamics of typical unimodal maps from the statistical point of view. We also describe the (generalized) renormalization...
Smooth Siegel disks via semicontinuity: a remark on a proof of Buff and Cheritat (2003)
Recently, Xavier Buff and Arnaud Cheritat have provided an elegant proof of the existence of quadratic Siegel disks with smooth boundary. In this short note, we show how results of Yoccoz and Risler...
Statistical properties of unimodal maps: the quadratic family (2003)
Avila, Artur, Moreira, Carlos Gustavo
We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps...
Statistical properties of unimodal maps: the quadratic family (2003)
Avila, Artur, Moreira, Carlos Gustavo
We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps...
Avila, Artur, Moreira, Carlos Gustavo
ICM2002 Satellite Conference: New Directions in Dynamical Systems Abstracts (August 2002 Kyoto)
Avila, Artur, Moreira, Carlos Gustavo
ICM2002 Satellite Conference: New Directions in Dynamical Systems Abstracts (August 2002 Kyoto)
Robust transitivity and topological mixing for $C^1$-flows (2002)
Abdenur, Flavio, Avila, Artur, Bochi, Jairo
We prove that non-trivial homoclinic classes of $C^r$-generic flows are topologically mixing. This implies that given $\Lambda$ a non-trivial $C^1$-robustly transitive set of a vector field $X$,...
Infinitesimal perturbations of rational maps (2001)
We analyze the infinitesimal effect of holomorphic perturbations of the dynamics of a structurally stable rational map on a neighborhood of its Julia set. This implies some restrictions on the...
Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative (2001)
Avila, Artur, Moreira, Carlos Gustavo
We prove that there is a residual set of families of smooth or analytic unimodal maps with quadratic critical point and negative Schwarzian derivative such that almost every non-regular parameter is...
Quasisymmetric robustness of the Collet-Eckmann condition in the quadratic family (2001)
Avila, Artur, Moreira, Carlos Gustavo
We consider quasisymmetric reparametrizations of the parameter space of the quadratic family. We prove that the set of quadratic maps which are either regular or Collet-Eckmann with polynomial...
A formula with some applications to the theory of Lyapunov exponents (2001)
We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub,...
Regular Or Stochastic Dynamics In Real Analytic Families Of Unimodal Maps (2001)
Artur Avila, Mikhail Lyubich, Welington De Melo
In this paper we prove that in any non-trivial real analytic family of unimodal maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely...
Statistical properties of unimodal maps: the quadratic family (2000)
Avila, Artur, Moreira, Carlos Gustavo
We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps...