62 research outputs found
On closed currents invariant by holomorphic foliations, I
We consider closed positive currents invariant by a singular holomorphic
foliation on an algebraic surface. We show that under some conditions the
foliation must leave invariant an algebraic curve
Discrete orbits, recurrence and solvable subgroups of Diff(C^2,0)
We discuss the local dynamics of a subgroup of Diff(C^2,0) possessing locally
discrete orbits as well as the structure of the recurrent set for more general
groups. It is proved, in particular, that a subgroup of Diff(C^2,0) possessing
locally discrete orbits must be virtually solvable. These results are of
considerable interest in problems concerning integrable systems.Comment: The first version of this paper and "A note on integrability and
finite orbits for subgroups of Diff(C^n,0)" are an expanded version of our
paper "Discrete orbits and special subgroups of Diff(C^n,0)". An intermediate
version re-submitted to the journal on March 2015 is available at
http://www.fep.up.pt/docentes/hreis/publications.htm where there is also a
comparison between these 3 version
Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlev\'e 6
We study the dynamics of the group of holomorphic automorphisms of the affine
cubic surfaces \begin{align*} S_{A,B,C,D} = \{(x,y,z) \in \mathbb{C}^3 \, : \,
x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D\}, \end{align*} where and are
complex parameters. We focus on a finite index subgroup whose action not only describes the dynamics of
Painlev\'e 6 differential equations but also arises naturally in the context of
character varieties. We define the Julia and Fatou sets of this group action
and prove that there is a dense orbit in the Julia set. In order to show that
the Julia set is ``large'' we consider a second dichotomy, between locally
discrete and locally non-discrete dynamics. For an open set in parameter space,
, we show that there simultaneously exists an
open set in on which acts locally discretely
and a second open set in on which acts locally
non-discretely. After removing a countable union of real-algebraic
hypersurfaces from we show that simultaneously
exhibits a non-empty Fatou set and also a Julia set having non-trivial
interior. The open set contains a natural family of parameters
previously studied by Dubrovin-Mazzocco.
The interplay between the Fatou/Julia dichotomy and the locally
discrete/non-discrete dichotomy plays a major theme in this paper and seems
bound to play an important role in further dynamical studies of holomorphic
automorphism groups.Comment: We have added further details on the connections between our work and
the dynamics of mapping class groups on character varieties, especially to
the Bowditch BQ conditions and the related works of several authors. We have
also made the writing more concise. Comments welcome
Questions about the dynamics on a natural family of affine cubic surfaces
We present several questions about the dynamics of the group of holomorphic
automorphisms of the affine cubic surfaces where
and are complex parameters. This group action describes the monodromy of
the famous Painlev\'e 6 Equation as well as the natural dynamics of the mapping
class group on the character varieties associated to
the once punctured torus and the four times punctured sphere.
The questions presented here arose while preparing our work ``Dynamics of
groups of automorphisms of character varieties and Fatou/Julia decomposition
for Painlev\'e~6'' \cite{RR} and during informal discussions with many people.
Several of the questions were posed at the Simons Symposium on Algebraic,
Complex and Arithmetic Dynamics that was held at Schloss Elmau, Germany, in
August 2022 as well as the MINT Summer School ``Facets of Complex Dynamics''
that was held in Toulouse, France, in June 2023.Comment: 11 pages, comments welcome
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