Parabolic implosion is a tool for studying the dynamics of perturbations of a map with a fixed point tangent to the identity, or more generally with at least one eigenvalue which is a root of unity. We will start by surveying classical parabolic implosion in dimension one, and then we will explain an ongoing work on parabolic implosion of germs tangent to the identity in dimension 2.
Joint...
In the 1980’s, William Thurston obtained his celebrated characterization of post-critically finite rational maps. This result laid the foundation of such a field as Thurston's theory in holomorphic dynamics, which has been actively developing in the last few decades. One of the most important problems in this area is the characterization question, which asks whether a given topological map is...
An important aspect of the theory of pseudo-Anosov mapping
classes concerns the study of the stretch factor lambda(f) of a
pseudo-Anosov mapping class f. This is a bi-Perron algebraic integer
of degree bounded above by 6g-6 which is the dimension of the
Teichmüller space for the underlying surface. The question of
realising any bi-Perron algebraic integer as a stretch factor is a
major...
We will look at the dynamics of a family of birational maps on R^2 and C^2. Then we will discuss a noninvertible rational map acting in dimension 2. This map arises as the renormalization map in the Iterated Monodromy group of the Basilica map.
We revisit this classical topic in the geometry of strata with an eye on arbitrary characteristic, using recent advances for compactifying strata
A translation surface is a surface endowed with
an atlas whose change of charts are translations. Fundamental
examples include the flat tori $\mathbb{C} / \Lambda$. A translation
surface comes with a one-parameter family of linear flows, one
for each direction in $\mathbb{C}$. Translation surfaces naturally
appear when considering billiard flows in rational polygons.
The main focus of...
In this talk we will present some results and research directions of John Hubbard which have been influential for the study of the dynamics of polynomial automorphisms of ${\mathbb C}^2$.
We discuss how to extend Thurston’s famous characterization theorem of rational maps to a natural and interesting class of transcendental maps. This is joint work work with Sergey Shemyakov and based on his PhD thesis, as well as extensions thereof.
