Short description of some works (new)
This concerns only my more recent work. Older stuff is
here.
The list below is not extensive, and I plan to address several
ommissions as I revise this page (particularly low-regularity cocycles,
Teichm\"uller and almost reducibility).
Let me call attention to one interesting development of my research,
which went much better than I had expected, and is described in more detail
in the ``Global theory'' papers (in the Schr\"odinger section below). The
dynamical interest in the study of quasiperiodic cocycles can be motivated
by the fact that it allows for both the KAM phenomenon of persistence of
quasiperiodic motions and for nonuniformly hyperbolic behavior.
My current work is dedicated to the precise understanding of the
transition from the
``persistently zero Lyapunov exponent'' regime to the positive Lyapunov
exponent as one moves through the parameter space, and in particular
establishes that one typically faces only finitely many phase
transitions. It is a key part in my (now completed) program of
establishing the Spectral Dichotomy (a precise description of the spectrum
of the associated operators), which I will describe in the near future.
On the regularization of conservative maps.
To appear in
Acta Mathematica.
It is well known that C^r maps can be ``regularized'', that is,
approximated by C^infty maps in the
C^r topology (r a positive integer). Such a result plays an important
role in differential topology, but also dynamical systems. Palis and Pugh
asked in the 1970's
whether the same result holds in the symplectic or volume-preserving
category. Zehnder soon showed that the answer was positive in the
symplectic case, and in the volume preserving case in high regularity (r at
least 2). It turned out that the low regularity volume preserving case is
genuinely special (the PDE's involved in this problem behave nastily
for r=1, as shown by Burago-Kleiner and McMullen in their study of
Gromov's question on the uniformization of separated nets in Euclidean
spaces). In this paper, we
show that C^1 volume preserving maps can neverthless be regularized.
Of course, C^1 is
also special in the theory of dynamical systems, since very strong
perturbation lemmas (Closing Lemma, Connecting Lemma) are only known in this
case. This result thus lifts a significant obstacle in the development of a
theory of C^1 generic volume preserving dynamics (in the introduction,
I comment on some consequences of regularization which became unconditional
with this work).
The full renormalization horseshor for unimodal maps of higher degree:
exponential contraction along hybrid classes.
Joint with M. Lyubich.
This paper is concerned with the
renormalization of analytic unimodal maps.
In our project to show that the full renormalization horseshoe is
uniformly hyperbolic for higher degree unimodal maps
(and from there to recover the statistical description of
the parameter space), the major
stumbling block was the proof of contraction of the
renormalization operator. The original approach to
this problem is Sullivan's, who put forward a program aiming at
the bounded type case, centered around a
basic precompactness property (``complex bounds''). This case was
eventually handled by McMullen, who proved exponential contraction
by a geometric technique ``McMullen towers''.
Later on, Lyubich proved exponential contraction
in degree 2 by a very detailed understanding
of the geometric possibilities: renormalization splits into two types
``essentially bounded'' and ``high'', where the first can be dealt
with by a (parabolic) tower technique,
and the second exhibts some obvious contracting features.
In higher degree, Lyubich's alternative no longer holds
and this approach is much more difficult to implement.
Our goal here is to give a proof of exponential convergence
that disregards the fine geometry and hence works equally
well in all cases. In fact, our proof is much simpler, taking
the focus out of the dynamics in the phase space
and moving it to the parameter space (in a sense
realizing Sullivan's original program):
exponential contraction is proved to be a ``soft'' consequence
of some of the features of the renormalization operator.
Those are basically: the existence of a holomorphic extension to
a ``hyperbolic-like'' parameter space, several precompactness
properties, and the simple topology of the parameter space.
Extremal Lyapunov exponents: an invariance principle and
applications.
Joint with Marcelo Viana.
To appear in Inventiones Mathematicae.
Cocycles over partially hyperbolic maps.
Joint with Jimmy Santamaria and Marcelo Viana.
Lyapunov exponents, absolute continuity and rigidity.
Joint with Marcelo Viana and Amie Wikinson.
This is a sequence of works about Lyapunov exponents of ``smooth cocycles''
fibering over a base dynamics with hyperbolic features and other dynamical systems which can be analyzed in terms of them.
The first two papers analyze the consequences of the vanishing of the
Lyapunov exponents (along the fibers), aiming particularly at showing
that it implies the existence of certain ``invariant sections'' (with
values at some appropriate bundle) with several invariance and
continuity properties (generalizing works of Furstenberg and
Ledrappier). To keep this short, instead of getting into a description
of the precise framework of those papers, I will just discuss a striking
application (joint with Marcelo Viana) which does
not need the introduction of too many definitions. Consider a genuine
``pseudo-Anosov'' automorphism of the 4d torus, i.e., it is a ergodic
symplectic automorphism whose eigenvalues intersect the unit circle. We
show that a symplectic perturbation is either nonuniformly hyperbolic
or volume preserving conjugate to the original automorphism. As a Corollary
we conclude that the original automorphism is stably Bernoulli (stable
ergodicity is already a deep result of Federico Rodriguez-Hertz, which does
play a role in our arguments).
The third paper develops the theory in a very different direction:
rigidity properties of invariant foliations. Perhaps the most surprising
result is the following.
A smooth perturbation of the time-one map of the geodesic flow on a
hyperbolic surface is a partially hyperbolic diffeomorphism of a 3
dimensional manifold, and has a one-dimensional center foliation
topologically the same as the orbit foliation of the original flow. We
show that if this foliation is absolutely continuous then the
perturbation embeds in a one-dimensional flow. On the other hand, if
this does not happen, we show that the center foliation is very far
from being absolutely continuous: the disintegration of Lebesgue
measure is atomic.
(Though Lyapunov exponents are not present at all in the above
formulation, the most difficult case to analyze is in fact when central
Lyapunov exponent is zero.)
Small eigenvalues of the Laplacian and mixing properties of
projective measures in moduli space.
Joint with Sébastien Gouëzel.
In this work, we study mixing properties of the Teichmuller flow with
respect to projective SL(2,R) invariant probability
measures in the moduli space (i.e., those measures
coming from volume forms on projective
varieties). Examples are given by the Masur-Veech measure on
strata which is of fundamental importance for the analysis of typical
translation surfaces, but the larger class of projective measures is
expected to play a determinant role in the
understanding of non-typical translation surfaces (including billiards):
it is widely conjectured that a Ratner-like phenomena takes place so that any
SL(2,R) orbit is ``equidistributed'' with respect to some projective measure.
As shown in (another) work of Ratner,
the complementary series component of the SL(2,R)
representation gives the main terms in the asymptotics of correlation
functions, hence this asymptotics is intimately connected with the spectrum
of the (foliated) Laplacian that lies in the interval (0,1/4). For the
Masur-Veech measure (in the case of strata of Abelian differentials),
we had previously shown (joint also with Yoccoz) the
spectral gap property, i.e., no spectrum near 0. This was also based on a
detailed combinatorial analysis centered on the Rauzy-Veech renormalization
algorithm, so it is really attached to the Masur-Veech measure (even
consideration of the Masur-Veech measure for strata of quadratic
differentials already creates complications, and was treated separately in a
paper with Resende). Moreover, there was no information going beyond the
spectral gap: in particular, the possibility of some
continuous spectrum, non-isolaled eigenvalues, or of eigenvalues of
infinite multiplicity, in (0,1/4),
remained (and either would prevent good asymptotics expressions for
correlation decay).
Here we prove the spectrum of the Laplacian in (0,1/4) consists entirely of
isolated eigenvalues of finite multiplicity, which can only accumulate at
1/4. As described above, this gives exponential mixing and good
asymptotics for the decay of correlations for projective measures.
Ergodic one-dimensional
Schr\"odinger operators and SL(2,R) cocycles.
My recent investigations on this topic were quite succesful, leading to
(positive or negative) answers to several key questions. They are
presented separately to avoid drowning my other research.
Global theory of one-frequency Schr\"odinger
operators I: stratified analyticity
of the Lyapunov exponent and the boundary of non-uniform hyperbolicity.
Global theory of one-frequency Schr\"odinger operators II: acriticality
and finiteness of phase transitions for typical potentials.
It is known that Schr\"odinger operators with one-frequency
analytic quasiperiodic potential behave distinctly for
small and large values of the coupling constant. Most progress
in the field so far has concerned with the understanding of
those two distinct regimes, but no authentic global theory
could be constructed since their interface has remained a mystery.
The goal of this series is precisely to describe the
phase transition. The highlights of the first paper are
the following. Energies in the spectrum can be separated
into three regimes, ``supercritical'',
``critical'' and ``subcritical''. Both supercritical and subcritical
regimes are stable. The critical regime is not (in fact it can be shown
to correspond to the boundary of the supercritical one in
the joint ``potential times energy'' parameter space, but this is delayed to the second paper because of a technicality).
The critical regime is contained in a countable union of
codimension-one submanifolds.
Besides the ``global theory'' aspect emphasized
above, the first paper has a secondary theme: the
study of regularity properties of the Lyapunov exponent.
It is known that the Lyapunov exponent can never be expected to
be more than H\"older, and often it is not even that. We show
that it has some other kind of very strong regularity
``stratified analyticity'', that is ideally suited to
describe the phase transition above. The source of this
regularity is a severe constraint on the behavior of the
Lyapunov exponent of SL(2,C) cocycles, ``quantization of
acceleration'', which we identify here. This quantization
implies that most SL(2,C) cocycles are ``regular'' (in the
sense that the acceleration is constant in a neighborhood).
Finally, we show that regularity is equivalent to uniform
hyperbolicity when the Lyapunov exponent is positive.
The goal of the second paper is to study the phase transition
for typical potentials. We show that the picture is as
good as could be possibly expected: there are only finitely
many alternances of regime. In fact, it is even better
than that: there are no critical energies at all in the
spectrum! Thus what we are really doing here
is showing that the different stable regimes are separated
by gaps in the spectrum.
Density of positive Lyapunov exponent for SL(2,R) cocycles.
One of the most basic problem in the theory of cocycles is whether a
cocycle with vanishing Lyapunov exponents may be approximated by one with
non-vanishing ones. Of course it makes sense to consider this question in
various regularity categories, and in higher regularity the approximation
task becomes increasingly difficult, since drastic perturbation are no
longer allowed. It is good to keep in mind that
somewhat similar questions about Lyapunov exponents
(for instance, whether it is possible to approximate non-vanishing
Lyapunov exponents by vanishing ones) do depend on regularity considerations.
In this paper we give a complete solution to this problem in the case of
SL(2,R) cocycles, for all
(usual) regularity classes (say, analytic, smooth, H\"older): positive
Lyapunov exponents are dense if and only if the base dynamics is not
periodic.
Absolute continuity without almost periodicity.
This is about the so-called Kotani-Last conjecture
for one-dimensional ergodic Schr\"odinger operators, which states that the
presence of some absolutely continuous spectrum must be almost periodic.
Absolutely continuous spectrum corresponds to the existence of positive
measure set of energies for which the Lyapunov exponent is zero, and I got
interested in this question after obtaining the results of
the previous paper.
This conjecture has been around for a while, and became a
central topic of the theory
after recent popularization (by Barry Simon,
Svetlana Jitomirskaya and David Damanik).
We produce a counterexample which is like a
``limit-periodic driven'' almost Mathieu operator.
To guarantee the existence of some absolutely
continuous spectrum, we work in the perturbative regime and do a
bit of KAM.
We also discuss the case of ergodic Schr\"odinger operators on the real line
(as opposed to the lattice), which is in fact much simpler to analyze. Here
we are able to produce a weak mixing example with purely ac spectrum,
which rules out the most natural ways to correct the Kotani-Last
conjecture...
Bulk universality and clock spacing of zeros for ergodic Jacobi matrices
with a.c. spectrum.
Joint with Barry Simon, Yoram Last.
Here we are interested in
the asymptotic distribution of eigenvalues of random matrices, in this case
Jacobi matrices generated by an ergodic process. Consider as usual finite
truncations of size N and let N grow.
It is known that eigenvalues have a non-random
asymptotic distribution (the integrated density of states), but here we are
interested on the local behavior: how the eigenvalues distribute in scale
1/N near some given point in the spectrum.
We obtain a complete answer in the region where the spectrum is
absolutely continuous: almost surely with respect to the ac component,
the eigenvalues distribute along arithmetic progressions.
At the time this
paper was produced, the only examples with ac spectrum were almost periodic:
thanks to the previous paper we now know that ac spectrum may arise for more
complicated dynamics as well.
A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies.
Joint with B. Fayad and R. Krikorian.
The basic problem of the local theory of Schr\"odinger cocycles is to study
whether such a cocycle is reducible (conjugate to a constant one). This was
first tackled by the KAM approach, in which one needs to keep two kinds of
small denominators under control: the ones related to the base frequency,
which are involved in the solution to the cohomological equation, and the
ones related to resonances between the fibered rotation number and the base
frequency. It is not surprising that one needs to impose a
Diophantine condition on the base frequency (reducibility of SO(2,R)
cocycles, a simpler problem,
is indeed equivalent to the analysis of the cohomological equation).
However, the most interesting aspect of SL(2,R) reducibility is really
whether one can solve the ``non-commutative part of the problem'', that is,
conjugate the cocycle to a SO(2,R) valued one. In this paper, we show that
for arbitrary irrational base frequencies, there is a positive probability
of ``reducibility to rotations'' for cocycles close to constant, a
surprising extension of the Dinaburg-Sinai result (which considered
Diophantine frequencies). The proof is based on
a ``non-standard'' KAM scheme which likes resonances on the base frequency.