Short description of some works
The discussion here is truncated around 2007.
For more recent stuff, see here.
Most of my initial work dealt with the dynamics of real
quadratic maps and more general unimodal maps. This naturally evolved
to
works on complex dynamics (Siegel disks, geometry of Julia sets).
Then I got quite interested on Lyapunov exponents, quasiperiodic
systems and the Schr\"odinger equation, and a bit later on
interval exchange transformations and Teichm\"uller flow.
A common feature of those three topics is the important role played by
renormalization algorithms (though much of my Schr\"odinger work is not
about renormalization).
Works on one-dimensional dynamics
My thesis (under the direction of Welington de Melo)
was on one-dimensional dynamics, and the real work started towards the end
of 1998.
Regular or stochastic dynamics in real analytic families of
unimodal maps.
Joint with Mikhail Lyubich and Welington de Melo.
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.
Joint with Carlos Gustavo Moreira.
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.
Joint with Carlos Gustavo Moreira.
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.
Joint with Xavier Buff and Arnaud Ch\'eritat.
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.
Joint with Mikhail Lyubich.
Journal of the American Mathematical Society 21 (2008), 305-363.
Examples of Feigenbaum Julia sets with small Hausdorff dimension.
Joint with Mikhail Lyubich.
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.
Joint with Jeremy Kahn, Mikhail Lyubich and Weixiao Shen.
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.
Joint with Mikhail Lyubich and Weixiao Shen.
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.
Joint with Carlos Gustavo Moreira.
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
My original motivation here was
to develop a global picture of the field from the point of view of
mainstream dynamical systems (my first work with Krikorian
in particular was an attempt to
obtain an analogous of Lyubich's regular or stochastic dychotomy for the
quadratic family). As I developed my interest in this topic, it became
clear that many of the natural questions did not translate to
problems of mainstream dynamical systems type. I went on to work on
very concrete questions, often running into problems
from Barry Simon's list of 15 problems on Schr\"odinger operators
for the 21st century (this list included three problems
on one-dimensional quasiperiodic operators, more
particularly on the model known as the almost Mathieu operator, that I will
discuss below). The better understanding obtained in the process eventually
allowed me to return to the ``global picture'' goal, about which I talk
elsewhere.
Reducibility or non-uniform hyperbolicity for quasiperiodic
Schr\"odinger cocycles.
Joint with Rapha\"el Krikorian.
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.
Joint with Svetlana Jitomirskaya.
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.
Joint with Svetlana Jitomirskaya.
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.
Joint with David Damanik.
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.
Joint with Rapha\"el Krikorian.
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
I started working on this in the end of 2003, though my interest had already
developed gradually with my contact with Yoccoz and Thouvenot.
It is again
a field where several different topics mix. Though probability is present
on all the
works below, there is also a heavy combinatorial aspect (in the second and
third papers), which is implemented to understand the boundary dynamics of
the Teichm\"uller flow. Let me comment a bit on that.
Intuitively, one can try to compactify the moduli
spaces by attaching a boundary corresponding to degenerate geometric
structures. Those actually correspond to moduli spaces of lower complexity:
degenerating a bitorus may yield a torus (with some markings).
Previous approaches are not very adapted to keeping track of
the dynamics of the Teichm\"uller flow. The one developed with Viana
describes how lower dimensional spaces attach to the boundary at the level
of the combinatorial model of the Teichm\"uller flow (the description
also preserves such information as the action of the dynamics on homology).
There we describe just enough cases for our purposes, but in the work with
Gou\"ezel and Yoccoz, we refine it to give optimal
estimates on the probability of
orbits staying "near the boundary" for a long time.
Weak mixing for interval exchange transformations and translation
flows.
Joint with Giovanni Forni.
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.
Joint with Marcelo Viana.
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.
Joint with Sebastien Gou\"ezel and
Jean-Christophe Yoccoz.
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).