Artur Avila

Works on one-dimensional dynamics

Regular or stochastic dynamics in real analytic families of unimodal maps.

Inventiones Mathematicae 154 (2003), 451-550.

Misha Lyubich showed that almost every real quadratic map was either regular (hyperbolic) or stochastic (has an absolutely continuous invariant probability). Here we generalize this dychotomy to the space of analytic unimodal maps. The actual main result (which is used to obtain the dychotomy, but also allows a detailed picture of the bifurcations in analytic families of unimodal maps) is to show that certain Banach spaces of analytic unimodal maps are laminated (up to annoying codimension-one phenomena) by topological conjugacy class, with analytic leaves and quasymmetric holonomy. There were several difficulties in this long paper, but I like especially the construction of the transverse direction to some topological conjugacy classes using ``localized'' analytic perturbations... This paper formed the first part of my thesis.

Statistical properties of unimodal maps: the quadratic family.

Annals of Mathematics 161 (2005), 831-881.

This one formed another part of my thesis. Here we prove that almost every real quadratic map is either regular or Collet-Eckmann (a condition which in this context is equivalent to having exponential decay of correlations for the absolutely continuous invariant probability). The parameter exclusion argument is complicated by the very rough (quasisymmetric) phase-parameter relation available in this context. We somehow compensate it by exploring the very fast (torrential) decreasing of scales along the renormalization algorithm.

Statistical properties of unimodal maps: physical measures, periodic points and pathological laminations.

Publications Math\'ematiques de l'IH\'ES 101 (2005), 1-67.

The method developed in the previous paper turned out to be very powerful (though somewhat difficult to implement), and we used it again in a considerably more complicated situation to compute an almost surely correct formula for the exponents of periodic orbits which only uses topological data as an input. The proof is done in the context of analytic unimodal maps, and the surprise is of course that exponents of periodic orbits can be varied independently in a fixed topological class, so the statement appears wrong, at first sight, for many people. The point is that Fubini theorem is not necessarily valid in this setting: the lamination in topological conjugacy classes is pathological. This paper also proves en passant that the critical orbit is equidistributed with respect to the absolutely continuous invariant probability, for typical analytic unimodal maps.

Siegel disks with smooth boundaries.

Acta Mathematica 193 (2004), 1-30.

Smooth Siegel disks via semicontinuity: a remark on a proof of Buff and Cheritat.

Those are about the existence of quadratic polynomials with a Siegel disk with smooth boundary (a problem connected with the existence of analytic diffeomorphisms of the circle that are smoothly, but not analytically, conjugated to a rotation). Non-quadratic smooth Siegel disks were first constructed by Perez-Marco (in an unpublished preprint), and it seems he could also adapt his method to cover the quadratic case. The point of this paper is to give a simple proof, and it started with an enlightening preprint of Buff and Ch�ritat which I later simplified in a short note (it was later merged with their preprint). The note I wrote makes it clear that Siegel disks with smooth boundary are a soft consequence of semicontinuity properties of the conformal radius of Siegel disks (hard proofs of which had been obtained in the quadratic case by Yoccoz and Riesler).

Hausdorff dimension and conformal measures of Feigenbaum Julia sets.

Journal of the American Mathematical Society 21 (2008), 305-363.

Examples of Feigenbaum Julia sets with small Hausdorff dimension.

In ``Dynamics on the Riemann Sphere. A Bodil Branner Festschrift'', Ed. Poul G. Hjorth, Carsten Lunde Petersen, European Mathematical Society (EMS), Z\"urich, pp. 71-87, 2006.

Next was my work with Lyubich on the Hausdorff dimension of Julia sets of Feigenbaum maps (infinitely renormalizable quadratic-like maps of bounded type and geometry, here I will discuss the case of a fixed point of renormalization). It was an open question (asked, for instance, in a book by McMullen) whether Feigenbaum Julia sets have always Hausdorff dimension $2$. This was suggested by analogy with Kleinian groups: indeed a common source of intuition in the field comes from trying to fill spots in ``Sullivan's dictionary'' between rational maps and Kleinian groups. We show that this intuition fails here.

Technically, our main achievement is to develop tools to estimate the critical exponent of the Poincar\'e series. The proof shows that the self-similarity of the Julia set translates approximately in a recurrence equation (which turns out to be quadratic) for the Poincar\'e series. The existence of positive fixed points for the equation implies convergence of the Poincar\'e series. We show how the coefficients of the equation are related to area estimates, which can be computed to sufficient precision in some cases to give examples of Feigenbaum Julia sets with Hausdorff dimension less than $2$. In general, we establish that there is in principle a trichotomy, with two ``stable regimes'' (from the point of view of decidability from finitely many estimates) and an intermediate unstable one. The stable regimes correspond to Julia sets of hyperbolic dimension less than 2, and differ on whether the Julia set has positive Lebesgue measure, while the intermediate regime has hyperbolic dimension 2 but a Julia set of positive Lebesgue measure. We found it interesting that the positivity of the Lebesgue measure of the Julia set forces the hyperbolic dimension to be less than the Hausdorff dimension (and I find it reasonable to expect that Feigenbaum Julia sets of positive Lebesgue measure exist). We wrote also a short version of this article which gives a specific sequence of examples which asymptotically achieves the trivial lower bound for the dimension.

Combinatorial rigidity for unicritical polynomials.

To appear in Annals of Mathematics.

This gives a simple proof of rigidity from apriori bounds (in the non-renormalizable case) that allows us to generalize Yoccoz's ``MLC (combinatorial rigidity) for non-renormalizable quadratic polynomials'' result to the higher degree case (the corresponding apriori bounds had been recently obtained by Kahn and Lyubich). The whole argument is very soft actually. Analytic expanding maps of the circle (of the same degree) are always quasisymmetric conjugate, and this can be shown by a standard ``pullback argument''. We find out that apriori bounds allows one to apply a pullback argument to the sequence of maps relating critical puzzle pieces of increasing levels, yielding uniform quasisymmetric estimates. This gives the (usually) most difficult step in the proof of rigidity.

Parapuzzle of the Multibrot set and typical dynamics of unimodal maps.

To appear in Journal of the European Mathematical Society.

Here we carry out the metric analysis of the parameter space of higher degree unicritical polynomials in the finitely renormalizable case. In the complex plane, we show that ``Yoccoz parameters'' (finitely renormalizable and without non-repelling periodic points) have zero Lebesgue measure, and even that they are not Lebesgue density point of the Multibrot set (higher degree analogous of the Mandelbrot set). In degree 2, this had been proved by Shishikura, but the higher degree case is much more interesting: while in degree 2 the Yoccoz parameters correspond to Julia sets of zero Lebesgue measure (and thus a natural phase-parameter transfer may be anticipated), this is a very unclear issue in higher degree. We also proceed the analysis of real parameter values, showing that the whole detailed description of the non-renormalizable parameters in the quadratic family goes through in the higher degree case as well.

Hausdorff dimension and the quadratic family.

Our original motivation in this work was to answer a question of Lyubich, of whether the set of infinitely renormalizable parameter values in the quadratic family had Hausdorff dimension less than one (Lyubich had shown measure zero). We went much further however, in two directions. On one hand, we showed that pathological behavior already appears with positive dimension: inexistence of physical measures, physical measures supported on expanding Cantor sets, non-ergodic physical measures. On the other hand, we show that the set of non-regular parameters which are not Collet-Eckmann has Hausdorff dimension less than one (thus, even in the sense of Hausdorff dimension, most parameters still do enjoy ``best possible'' statistical description). This result follows a very different path than the ``measure zero'' result that was earlier obtained, since the statistical analysis routinely gets rid of sets of Hausdorff dimension one. Matters are indeed much more complicated here, since exponential recurrence of the critical orbit, for instance, happens for a set of parameters of Hausdorff dimension one. Our proof implements a very robust parameter exclusion process, estimating the ``multifractal spectrum of hyperbolicity'', which allows us to target the parameter exclusion process to be more strict on regions which are weakly hyperbolic.

Works on the Schr\"odinger equation and quasiperiodic cocycles

elsewhere

Reducibility or non-uniform hyperbolicity for quasiperiodic Schr\"odinger cocycles.

Annals of Mathematics 164 (2006), 911-940.

My first work in the field. We proved that almost every analytic Schr\"odinger cocycle over an irrational rotation either has a positive Lyapunov exponent or is conjugate to an eliptic constant cocycle. Our motivation was mainly dynamical, but as an immediate corollary we could solve one of the problems in Barry Simon's list: the spectrum of the almost Mathieu operator at the critical point has Lebesgue measure zero, for all frequencies. The proof was based on renormalization, and used ideology from one-dimensional dynamics (prove apriori bounds, then look at the limits,...). Since we could exploit KAM theory near the limits, this was a very soft paper, except for the proof of apriori bounds, as usual.

The Ten Martini Problem.

To appear in Annals of Mathematics.

The Ten Martini Problem asks one to show that the spectrum of the almost Mathieu operator is a Cantor set for all couplings and frequencies (another problem in Barry Simon's list). In dynamical systems terms, one should show density of hyperbolicity in certain families of dynamical systems. The proof has both soft and hard parts. In the soft part, we argue by contradiction exploring the consequences of an interval in the spectrum. We show that we can essentially bring the problem down from SL(2,R) to SO(2,R), by applying basic complex analysis to strengthen a result of Kotani (it was very surprising that this idea had never been used here). We also use Hartogs type arguments (subharmonic functions), so that we can exclude annoying parameters from the analysis. In the end, one still has to solve some explicit cohomological equation for enough values of the parameter. This is done by hand (no contradiction argument this time), via a sophisticated proof of localization for some difficult values of the parameter. There are still some remaining value of the parameter, which we can take care of by Liouville type arguments (a strengthening of the estimates of Choi-Elliot-Yui). Let us just say that we were very lucky to manage to cover the whole parameter space with those different arguments.

Almost localization and almost reducibility.

To appear in Journal of the European Mathematical Society.

Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling.

Inventiones Mathematicae 172 (2008), 439-453.

Absolutely continuous spectrum for the almost Mathieu operator.

The next problem in Barry Simon's list: was to show that for subcritical coupling the spectral measures are always absolutely continuous. My first contact with this problem came through a ramification of my work with Jitomirskaya on Schr\"odinger operators with Diophantine frequency and small analytic potential. We worked out a quantitative link between localization-type estimates (for the dual model) and reducibility-type estimates: in a sense, we managed to control ``conjugacies'' which may not really exist, through their possibly diverging Fourier series and dynamical analysis. Among several other results, about which I will try to write later, this allowed us to conclude pure ac spectrum for Diophantine frequencies.

At this point I decided to work more directly on the full Simon's question. Initially the situation was not looking very good from the Liouville side, but this changed with my work with Damanik (on the exponentially Liouville case): we found out a completely different approach that could control the averages of spectral measures (that is, the integrated density of the states) by dynamical analysis of perturbation of periodic approximants (and using lots of explicit computations known for the almost Mathieu operator...).

The full result still seemed elusive, particularly from the exponentially Liouville side, since the understanding of the dynamics was still very weak (while averaging is good for almost sure results, the full result is related to the dynamics for individual phases). I eventually bypassed this problem, developing a new cancelation argument for the phase parameter, and proceeded to complete the proof by strengthening the Diophantine side (estimates for the Corona problem are helpful here, though not essential), in my recent preprint.

Monotonic cocycles.

The theory of quasiperiodic SL(2,R) cocycles is much more well understood in the case of cocycles homotopic to a constant (due primarily to the fact that Schr\"odinger cocycles fall into this case). In this work we demonstrate that the non-homotopic to the constant theory (in any number of frequencies) presents remarkable features that are at odds with what we came to expect from the analysis of the Schr\"odinger equation. Our key achievement is the development of a ``local theory'' in this context. In the Schr\"odinger case, local means traditionally ``close to constant'', but can be equivalently formulated as ``close to rotations'' (since SO(2,R) cocycles can be always brought close to constant), and this second formulation still makes sense in the non-homotopic to constant case.

The traditional local analysis of Schr\"odinger cocycles is KAM based, and even more recent approaches (see works above) rely on the detailed analysis of the ``small denominators'': this is unavoidable, since the small denominators do interfere qualitatively with the dynamics. The analysis is also very sensitive to smoothness requirements (changing from analytic to Gevrey class already has qualitative consequences). But in this setting we are able to apply a completely different approach, and it turns out that the theory we obtain is much more robust.

We prove smoothness of the Lyapunov exponent (no better than 1/2-Holder can be obtained for the Schrodinger equation, and only under constraints on the small denominators), a sharp rigidity result (zero Lyapunov exponent implies smooth conjugacy to rotations), and minimality of the projective action. In the one-frequency case, a stronger ``convergence of renormalization'' result (compared to the first paper with Krikorian) is also proved, leading to global results.

Works on interval exchange transformations and Teichm\"uller flow

Weak mixing for interval exchange transformations and translation flows.

Annals of Mathematics 165 (2007), 637-664.

My first work in the field. It proves that translation flows in higher genus, and the associated interval exchange transformations, are almost surely weak mixing, that is, there are no eigenvalues. Essentially, this means showing that those systems do not factor measurably over a rotation. This question had been around since the 70's. We tackle the issue by focusing heavily in the parameter exclusion process. We first found out that, if allowed to do ``linear'' parameter exclusion, each positive Lyapunov exponent for the Kontsevich-Zorich cocycle (see below) gives an obstruction to the eigenvalue equation. Since the equation has one free parameter, two positive exponents should be enough. This only uses the ergodicity of the Teichm\"uller flow, and it is enough for the case of translation flows. But the case of interval exchange maps has less parameters and we can not rely on linear exclusion. A much more complicated argument is developed, using fully the chaoticity of the renormalization operator (successive renormalizations are essentially independent random variables), which allows us to efficiently model the parameter exclusion probabilistically. The probabilistic model is novel, so most of the effort is dedicated to its analysis. Those are some of my favorite estimates. A simple, but crucial detail in the approach taken here (and in the next two works) is that the benefit of working with a stochastic model (with ``independence'' or ``bounded distortion'') is worth enduring quite a lot of complications (like dealing with unbounded matrices).

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

Acta Mathematica 198 (2007), 1-56.

Next I worked with Viana in the proof of the Kontsevich-Zorich conjecture. The work of Zorich and Kontsevich had quantitatively linked the Lyapunov spectrum of the Zorich cocycle to the homological deviation of the orbits of a typical translation flow. At the level of translation surfaces, this cocycle corresponds to the Kontsevich-Zorich cocycle over the Teichm\"uller flow (on connected components of stratas of the moduli space of abelian differentias). The conjecture that the Lyapunov spectrum of this cocycle was simple was supported by numerical evidence. Zorich had shown (using Veech's work) that the extremal Lyapunov exponents are simple, and Giovanni had shown that the middle ones are distinct (due to symmetry, this is equivalent to all exponents being non-zero). The proof uses again chaoticity of the renormalization operator, this time to model the cocycle as a random matrix product in such way that a combinatorial criterium for simplicity of the spectrum (in the line of the random works of Guivarch-Raugy and Goldsheid-Margulis) can be derived. Verification of the combinatorial criterium is by induction on the complexity. This is inspired by the traditional procedure of ``going to the boundary of moduli space'', but is done in a novel way to keep track, combinatorially, of the dynamics.

Exponential mixing for the Teichm\"uller flow.

Publications Math\'ematiques de l'IH\'ES 104 (2006), 143-211.

In the previous works I used a lot chaoticity of the renormalization operator, which is very easy to prove (once one accepts the price of unbounded matrices). What about chaoticity of the Teichm\"uller flow itself? This is the topic of the IH\'ES paper with Gou\"ezel and Yoccoz, which shows that the Teichm\"uller flow is exponentially mixing in the moduli space of Abelian differentials. In order to apply the approach of Dolgopyat and Baladi-Vall\'ee to exponential mixing, we look at the Teichm\"uller flow (or some cover of it) as suspension over the renormalization algorithm. The crucial issue here is that this time we need good estimates on the size of the matrices (related to the return times). We need to investigate the probability of staying a long time ``close to the boundary''. The approach we implement (that leads to an optimal estimate) uses as combinatorial framework the induction on the complexity developed in the previous work (in a more refined form).

Short description of some works

Works on one-dimensional dynamics

Works on the Schr\"odinger equation and quasiperiodic cocycles

Works on interval exchange transformations and Teichm\"uller flow