Charles Favre: Recherche
 
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My research has been recently focused on the following four subjects:

Potential theory on Berkovich spaces Singularities of psh function Holomorphic dynamics Nonarchimedean dynamics

Here is list of my collaborators.



Potential theory on Berkovich spaces

Berkovich theory was introduced in the late 80's by Berkovich. It gives a natural way to study analytic geometry over non-archimedean fields. In this theory, analytic spaces (usually referred to as Berkovich spaces) have the great advantage of being both arcwise connected and locally compact. We refer to Ducros' webpage for a good introduction to the subject and for an extensive bibliography on its developements and applications.

What is potential theory on Berkovich spaces? Berkovich spaces being the analogs of complex analytic spaces over non-archimedean fields, it is natural to trasnport complex tools to the non-archimedean setting. Among these tools, the analysis revolving around plurisubharmonic functions, positive currents, and the ddc operator plays a particularly important role (see Demailly's webpage where accessible surveys on the subject can be freely downloaded). It thus seems natural to extend this pluripotential analysis to the non-archimedean context. Beside this fundamental aspect, pluripotential theory on Berkovich spaces has potentially numerous applications: in arithmetics (see Chambert-Loir's preprints on his webpage); in the study of singularities of psh function; in holomorphic and non-archimedean dynamics.

The one-dimensional case, i.e. the case of Berkovich curve, is now well understood, thanks to the joint efforts of Baker-Rumely, Favre-Jonsson, and Thuillier. We refer to the PhD  thesis of the last named author for a general construction of a Laplace operator on Berkovich curves; and to our joint book with M. Jonsson The Valuative Tree for a construction of a Laplace operator on a general metrized tree. The main reason why the one-dimensional case is easier to understand lies in the fact that locally any Berkovich curve looks like a tree. Below is an artistic view (done by M. Jonsson) of a neighborhood of a point in a Berkovich curve. Typically the boundary of such a neighborhood consists of finitely many points.
The Laplace operator is then a kind of mixture of the standard laplacian - d2/dx2 on the real line - which takes into account the "continuous" part of the tree -, and the laplacian on a finite graph defined in a combinatorial way - which takes into account the behaviour at the branching points- .

a Berkovich curve

In higher dimension, one should think of a local neighborhood of a Berkovich space as a union of (affine) simplices patched together in such a way that the resulting space remains contractible. A more mathematical formulation of this image is given by Kontesevich-Soibelman, and also in a recent paper of Thuiller. Although the affine structure on each simplex allows one to build a natural Monge-Ampere operator, it seems complicated to patch all these together. To overcom this difficulty, one is lead to a more geometric construction of the Laplace operator.
The key idea is to realize that the Laplace operator (and more generally the potential theory) is the incarnation of the intersection theory of divisors on all reductions of the Berkovich space. A construction of a Monge-Ampere operator on a Berkovich-like space (with a view towards application in singularities) is explained in my joint paper Valuations and plurisubharmonic singularities with S. Boucksom and M. Jonsson.

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Singularities of psh function                                                                              

 A plurisubharmonic function (or psh function in short) is a real-valued function defined on a smooth complex space whose convexity properties are adapted to the complex structure of the space. A singularity of  such a function is a point where  the value becomes negative infinite. Singularities of psh function are particularly important as being a source of construction of analytic objects, and as obstructions in the intersection theory of positive closed current. We refer to the survey Plurisubharmonic functions and potential theory in several complex variables by Kiselman for more informations on the history of the subject.
Various numbers have been introduced to measure the singularity of a psh function by Lelong and subsequently by Demailly and Kiselman. For instance, the Lelong number at 0 of the psh function log|f| with f holomorphic is equal to the multiplicity of the hypersurface {f=0} at 0. In the aforementioned paper Valuations and plurisubharmonic singularities written with S. Boucksom and M. Jonsson, we have shown that many of the fine properties of a psh function near a singular point 0 can be understood through the data of all its Lelong number at all infinitely near points (that is points lying on some exceptional divisor of a sequence of blowups above 0).
 
Using potential theory on Berkovich curves, we were also able with M. Jonsson to settle positively the following two questions in dimension 2:

We refer to our paper Valuations and plurisubharmonic singularities for an approach to these questions in higher dimension.

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Holomorphic dynamics

Holomorphic dynamics concerns the iteration of holomorphic maps. There is a huge literature on the subject in one dimension (a series of surveys on the subject can be downloaded from Stony Brook). In higher dimension, the theory is much more recent and uses quite different techniques from its one-dimensional analog. I am working on the more algebraic aspects of the theory and more precisely on the problem of description of the growth of degrees. For  a fixed rational map F: Cn  → Cn, define the degree deg(F) as the maximum of the degrees of its coordinates.

Problem.   Describe the behaviour of the sequence deg(Fk) when k tends to infinity.

One can make this problem more precise as follows. Note first that deg(Fk+l) ≤ deg(Fk) deg(Fl) hence λ(F) =  limk → ∞ deg(Fk)1/k  is well-defined. It is called the asymptotic degree of F (or first dynamical degree), and is invariant under conjugacy by a birational map.

Question 1: for any rational map, the asymptotic degree is an algebraic integer.

Question 2:  deg(Fk) = c. λ(F)k + O(hk) with h <λ(F) and c>0,  except if  F preserves an algebraic fibration.


With J. Diller, we have settled positively these two questions for birational maps in dimension 2. What we did was finding a good birational model in which the action of F on the cohomology became compatible with the iteration.

In the case of polynomial maps of C2, we have also settled these questions with S. Boucksom and M. Jonsson in Eigenvaluation and Degree growth for meromorphic surface maps, and have made substantial progresses  for general rational surface maps. The main difference with the invertible case is that in general no good birational model exists where the action of F on the cohomology is analyzable. We thus have to deal with all birational models at the same time, and define and study a good cohomology theory on the space one obtains after blowing up all points over P2.
We also refer to S. Cantat's paper Groupes de transformations birationnels du plan
for another applications of this kind of ideas.



In higher dimension though the situation is completely open, and the two questions above remain a great challenge. Viet-Anh Nguyen has introduced the class of quasi-algebraically stable maps for which the analysis of growth of degrees is easy. And
there are series of interesting examples of birational maps acting on the space of complex square matrices which were worked out numerically by a group of physicists leaded by J-M. Maillard, and theoretically (see also) by E. Bedford and K. Kim. But a general theoretical approach remains to be built.

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Nonarchimedean dynamics

The basic problem is to study the dynamics of rational maps R ∈ K(T) where K is a field endowed with a non-archimedean norm. We usually assume K to be complete and algebraically closed but this is not the only interesting case, see Benedetto-Briend-Perdry's paper for instance.
Benedetto and J. Rivera-Letelier have extensively developped the analog of the complex Fatou-Julia theory in this context whose feature is as follows.

Because the standard projective line over K is both totally discontinuous and (at least when the residue field of K is uncoutable) not locally compact, one is naturally led to look at the action of R on the Berkovich projective line P1ber(K). This space is  a compact tree (see the picture above). One can then split the Berkovich line into two totally invariant sets on which the dynamics of R exhibits completely different behaviours.

On the Fatou set (which is open), the dynamics is regular. This means that there is a good understanding of the dynamics on the preperiodic connected components of the Fatou set (see Rivera-Letelier's papers) ; and there are reasonable conjectures concerning the appearance of wandering components,  as the following paper  of Benedetto indicates. Note that the situation is here very different from the complex one, where Sullivan's non-wandering theorem states that all connected components of the Fatou set are necessarily preperiodic.

On the Julia set (which is compact), the dynamics is essentially chaotic. In order to describe it, it is thus natural to use statistichal tools and ergodic theory. As a first step we have constructed with Rivera-Letelier a mixing measure whose support coincides with the Julia set, see . This measure is not of maximal entropy as in the complex case in general, and it is a very delicate task to compute explicitely the entropy of it. However one can give bounds on the metric (and topological) entropy, and completely characterize  those rational maps with zero topological entropy: these are rational maps with good reduction (possibly after a conjugation by a Mobius transformation). In this case, their Julia set is reduced to a point.


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Collaborators

       Sebastien Boucksom       Jan Kiwi
      Serge Cantat       Luis Gustavo Mendes
       Jeffrey Diller 
      Jorge Pereira       
       Romain Dujardin               Juan Rivera-Letelier
       Vincent Guedj            Eugenio Trucco
       Mattias Jonsson       Elizabeth Wulcan


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