to itself
given by 
, where 
is a
complex polynomial and 
(this parameter is
actually the jacobian of the map).
I am interested in the iteration of Complex Hénon mappings.
They are biholomorphic mappings from to itself
given by , where is a
complex polynomial and (this parameter is
actually the jacobian of the map).
In my article [2] (see List of publications), I gave a topological model completely describing the
dynamics when 
has an attracting fixed point and
is small.
Theorem 1. For any 
in the main
cardioid 
of the Mandelbrot set, there exists
an 
such that:
for any 
satisfying 
,
there exists a homeomorphism 
which conjugates
to 
.
The map 
of the model 
is
given by the following recipe:
Consider T the solid torus 
, and
notice that the map 
defined by
can be extended to a homeomorphism 
of the whole 3-sphere 
. Now contains two invariant solenoids, 
(obtained by forward and backward iteration of 
).
,
where 
using polar coordinates 
in 
.
of the model:
.
I am pursuing actively this line of research with the aim of giving similar topological descriptions of maps that are not perturbations. The methods I use are both analytical (holomorphic motions, puzzles, laminations,...) and topological (Smale's theorem).
The escaping set for a complex Hénon mapping is the set of points with
bounded forward orbit. It has the fascinating property of having a very
complicated fractal boundary, and at the same time a rather rigid
complex structure (it has a universal cover biholomorphic to 
). I proved that in the case where the polynomial 
is quadratic, the Hénon map itself is prescribed by the
analytic structure of the escaping set.
More precisely I proved:
Theorem 2.
I have been trying to use the knowledge of these various analytic structures to produce interesting complex surfaces, by quotienting these escaping sets by well-chosen groups of automorphisms.
The punctured solenoid 
is an inverse
limit of the system of all finite unbranched covers of a punctured
surface of negative Euler characteristic.
The baseleaf preserving mapping class group 
consists of all homotopy classes of self maps of the solenoid which
preserve a certain path component that we call the distinguished leaf.
Together with D.Saric and R.C. Penner , in the article [1] (see List of publications) we gave a presentation for , using its action on a certain Triangulation
complex whose vertices are ideal triangulations of the unit disk
invariant under finite index subgroups of 
(see
the article for the description of the edges and the two-cells of the
complex).
Theorem 3. The triangulation complex is connected and simply
connected. The action of 
on this complex is
cellular.
Using this cellular action, we managed to obtain a presentation
for inspired by the Bass-Serre theory
of groups acting on trees. See [BPS] for the description (which needs
longer preliminary definitions).
I have been recently working on a certain class of ramified covering
maps from the 2-sphere 
to itself known as
Thurston mappings : these are orientation-preserving ramified
covering maps 
such that all the ramification
points have finite orbits.
A lot is already known about these but my recent goal has been to study these maps using only topological and combinatorial methods (vs. Teichmüller theory). This is a work in progress.