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Artur Avila

Publications and source records attributed to Artur Avila.

At least 37 records · Page 2Linked to original sources
arXiv2019

Symplectomorphisms with positive metric entropy

We obtain a dichotomy for $C^1$-generic symplectomorphisms: either all the Lyapunov exponents of almost every point vanish, or the map is partially hyperbolic and ergodic with respect to volume. This completes a program first put forth by Ricardo Mañé. A main ingredient in our proof is a generalization to partially hyperbolic invariant sets of the main result in [Dolgopyat-Wilkinson] that stable accessibility is $C^1$ dense among partially hyperbolic diffeomorphisms.

math.DS↗
arXiv2018

Livsic theorem for diffeomorphism cocycles

We prove the so called Livšic theorem for cocycles taking values in the group of $C^{1+β}-diffeomorphisms of any closed manifold of arbitrary dimension. Since no localization hypothesis is assumed, this result is completely global in the space of cocycles and thus extends the previous result of the second author and Potrie [KP16] to higher dimensions.

math.DS↗
arXiv2017

Diffeomorphisms with positive metric entropy

We obtain a dichotomy for $C^1$-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.

math.DS↗
arXiv2017

Zorich conjecture for hyperelliptic Rauzy-Veech groups

We describe the structure of hyperelliptic Rauzy diagrams and hyperelliptic Rauzy-Veech groups. In particular, this provides a solution of the hyperelliptic cases of a conjecture of Zorich on the Zariski closure of Rauzy-Veech groups.

math.DS↗
arXiv2017

Complex one-frequency cocycles

We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other hand, this last property provides a good control of the periodic approximations of a cocycle, allowing us to show that domination can be characterized, in the presence of a gap in the Lyapunov spectrum, by additional regularity of the dependence of sums of Lyapunov exponents.

math.DS↗
arXiv2017

Weak mixing properties of interval exchange transformations and translation flows

Let $d >1$. In this paper we show that for an irreducible permutation $π$ which is not a rotation, the set of $[λ]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([λ],π)$ is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates on the Hausdorff dimension of the set of "bad" parameters in the context of fast decaying cocycles, following a strategy developed by V. Delecroix and the first author.

math.DS↗
arXiv2016

Second Phase transition line

We study the phase transion line of the almost Mathieu operator, that separates arithmetic regions corresponding to singular continuous and a.e. pure point regimes, and prove that both purely singular continuous and a.e. pure point spectrum occur for dense sets of frequencies.

math-ph↗
arXiv2016

Diffusion for chaotic plane sections of 3-periodic surfaces

We study chaotic plane sections of some particular family of triply periodic surfaces. The question about possible behavior of such sections was posed by S. P. Novikov. We prove some estimations on the diffusion rate of these sections using the connection between Novikov's problem and systems of isometries - some natural generalization of interval exchange transformations. Using thermodynamical formalism, we construct an invariant measure for systems of isometries of a special class called the Rauzy gasket, and investigate the main properties of the Lyapunov spectrum of the corresponding suspension flow.

math.DS↗
arXiv2015

Sharp Phase transitions for the almost Mathieu operator

It is known that the spectral type of the almost Mathieu operator depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya's conjecture in \cite{Ji95,J07}. Together with \cite{Aab}, we give the sharp description of phase transitions for the almost Mathieu operator.

math.SP↗
arXiv2015

Weak mixing directions in non-arithmetic Veech surfaces

We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square and hexagon). More generally, we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non-arithmetic Veech surface, and show that the Hausdorff dimension of the set of non-weak mixing directions is not full. We also provide a necessary condition, verified for instance by the Veech surface corresponding to the billiard in the pentagon, for the set of non-weak mixing directions to have positive Hausdorff dimension.

math.DS↗
arXiv2015

Some monoids of Pisot matrices

We show that several monoids of non-negative integer matrices enjoy a Pisot property: each matrix in that monoid has only one eigenvalue with absolute value larger than one. These monoids come from multidimensional continued fractions, namely the fully subtractive and Brun continued fractions.

math.DS↗
arXiv2015

On the Hausdorff dimension of the Rauzy gasket

In this paper, we prove that the Hausdorff dimension of the Rauzy gasket is less than 2. By this result, we answer a question addressed by Pierre Arnoux. Also, this question is a very particular case of the conjecture stated by S.P. Novikov and A. Ya. Maltsev in 2003.

math.DS↗
arXiv2015

Lebesgue measure of Feigenbaum Julia sets

We construct Feigenbaum quadratic polynomials whose Julia sets have positive Lebesgue measure. They provide first examples of rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set. The corresponding set of parameters has positive Hausdorff dimension.

math.DS↗
arXiv2014

Symplectic and Isometric SL(2,R) invariant subbundles of the Hodge bundle

Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N. As a corollary, it follows that the Hodge bundle of N is semisimple.

math.DS↗
arXiv2014

Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators

Simon's subshift conjecture states that for every aperiodic minimal subshift of Verblunsky coefficients, the common essential support of the associated measures has zero Lebesgue measure. We disprove this conjecture in this paper, both in the form stated and in the analogous formulation of it for discrete Schrödinger operators. In addition we prove a weak version of the conjecture in the Schrödinger setting. Namely, under some additional assumptions on the subshift, we show that the density of states measure, a natural measure associated with the operator family and whose topological support is equal to the spectrum, is singular. We also consider one-frequency quasi-periodic Schrödinger operators with continuous sampling functions and show that generically, the density of states measure is singular as well.

math.SP↗