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Artur Avila

Publications and source records attributed to Artur Avila.

At least 73 records · Page 4Linked to original sources
arXiv2008

Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum

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 $\frac{1}{n} K_n(x,x)$ for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. We then show that these hypotheses are always valid for ergodic Jacobi matrices with a.c. spectrum and prove that the limit of $\frac{1}{n} K_n(x,x)$ is $ρ_\infty(x)/w(x)$ where $ρ_\infty$ is the density of zeros and $w$ is the a.c. weight of the spectral measure.

math.SP↗
arXiv2008

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

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 that its iterates can be written as short compositions involving finitely many smooth diffeomorphisms). Thus rotations are distortion elements.

math.DS↗
arXiv2008

Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles

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 also consider the more general situation where the allowed products of matrices satisfy a Markovian rule.

math.DS↗
arXiv2008

On the spectrum and Lyapunov exponent of limit periodic Schrodinger operators

We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schrödinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero Lebesgue measure. No example with those properties was previously known, even in the larger class of ergodic potentials. We also conclude that the generic limit periodic potential has a spectrum of zero Lebesgue measure.

math.DS↗
arXiv2008

Almost localization and almost reducibility

We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schrödinger cocycles associated to a non-perturbatively small analytic potential and Diophantine frequency. In particular, we establish the full version of Eliasson's reducibility theory in this regime (our approach actually leads to improvements even in the perturbative regime: we are able to show, for all energies, ``almost reducibility'' in some band of analyticity). We also prove 1/2-Hölder continuity of the integrated density of states. For the almost Mathieu operator, our results hold through the entire regime of sub-critical coupling and imply also the dry version of the Ten Martini Problem for the concerned parameters.

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arXiv2008

Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts

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 which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrödinger operator is a Cantor set.

math.DS↗
arXiv2008

Parapuzzle of the Multibrot set and typical dynamics of unimodal maps

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 renormalizable. For real parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic, or Collet-Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the ``principal nest'' of parapuzzle pieces.

math.DS↗
arXiv2007

Absolute Continuity of the Integrated Density of States for the Almost Mathieu Operator with Non-Critical Coupling

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 is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schrödinger operator problems for the twenty-first century.

math.DS↗
arXiv2007

Reducibility or non-uniform hyperbolicity for quasiperiodic Schrodinger cocycles

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 either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrodinger operator, and allows us to complete the proof of the Aubry-Andre conjecture on the measure of the spectrum of the Almost Mathieu Operator.

math.DS↗
arXiv2006

A generic $C^1$ map has no absolutely continuous invariant probability measure

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 respect to Lebesgue) invariant probability measure. We show that this is a residual (dense $G_δ) set in the $C^1$ topology. In the course of the proof, we need a generalization of the usual Rokhlin tower lemma to non-invariant measures. That result may be of independent interest.

math.DS↗
arXiv2006

Simplicity of Lyapunov spectra: a sufficient criterion

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, who considered products of random matrices, and of Bonatti-Viana, who dealt with the case when the base dynamics is a subshift of finite type. Here the Markov structure may have infinitely many symbols and the ambient space needs not be compact. As an application, in another paper we prove the Zorich-Kontsevich conjecture on the Lyapunov spectrum of the Teichmüller flow in the space of translation surfaces.

math.DS↗
arXiv2005

Exponential mixing for the Teichmuller flow

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 (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Holder observables. A geometric consequence is that the $\SL(2,\R)$ action in the moduli space has a spectral gap.

math.DS↗
arXiv2005

Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture

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 differentials on compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface.

math.DS↗