You may have heard about the Mandelbrot set, a collection of “non-escaping” parameters in the moduli space of quadratic polynomials. Since every quadratic rational map has two critical points, we can think about the Mandelbrot set as living inside the collection of quadratic rational maps where one critical point is fixed, or \(Per_1(0).\) What if we instead looked at the quadratic rational maps where one critical point was \(n\)-periodic? Doing so yields the family of \(Per_n(0\)) curves whose structure is not yet well-understood, although looking at the pictures there are Mandelbrot-like sets living inside. In this talk, we detail these \(Per_n(0)\) curves and present how understanding the \(n\)-periodic polynomials in the Mandelbrot set can shed light on the topology and geometry of \(Per_n(0).\)

You By definition, the Mandelbrot set \(M\) is the set of all complex numbers c for which the orbit of \(0\) is bounded under iteration of \(f(z) = z^2 + c.\) Within M, there is a distinguished subset of what we call PCF parameters, where the orbit of \(0\) is finite. These parameters are special from both a dynamical and – somewhat surprisingly – a number-theoretic point of view. In this talk, we’ll explore how the geometry of these PCF parameters is restricted by the number theory. This is joint work with Myrto Mavraki.

For \(c\in C^*\), we consider the parametric family, Bergweiler’s family
$$f_c(z) = c-\log c + 2z – e^z$$
Since \(f_c\) respects the equivalence relation defined by \(z\sim w\) if \(z-w\in 2\pi i Z\), it induces a map on the quotient space \(Q=C/\sim\)
$$F_c:Q\to Q.$$
In this talk we study the properties of quasi-conformal conjugacies of the family \(f_c\) through the quotient map \(F_c\). In particular, we are interested in the case \(c\in D(2,1)\) which is the main hyperbolic component of the parameter space.

Understanding the structure of the Mandelbrot set has been a fundamental driver in the holomorphic dynamics. Near parabolic parameters certain repeating patterns called Elephants arise. In this talk we will discuss recent progresses in describing these elephants, and applications to understanding the geometry of the Mandelbrot set.

In the complex two-dimensional parameter space of quadratic rational maps, it is of interest to study dynamically defined one-dimensional slices. An interesting collection of such slices are the curves formed by maps having a periodic critical point, of a given period. We obtain a formula for the Euler Characteristic of these curves. The formula stems from the study of degenerate holomorphic families of quadratic rational maps and their rescaling limits.

Through the theory of quasiconformal surgery, Shishikura constructed in the late 1980s examples of rational maps of degree \(d > 2\) with cycles of Herman rings (HR) and provided a sharp upper bound for the number of these cycles. A consequence of this bound is that quadratic rational maps cannot have HR. In this talk I will discuss the realization of Herman rings by elliptic functions of order \( d > 2,\) and in particular, establish that any order \(2\) elliptic function cannot have cycles of Herman rings, generalizing a previous result by Hawkins & Koss for the Weierstrass P function. Then, I will present an upper bound for the number of invariant Herman rings and explain how to refine this bound in terms of the multiplicity of poles, regardless of the geometry of the lattice associated with the elliptic function.

Given a real vector field in dimension \(n\), a fundamental question is whether it is completely integrable, either locally or globally. This means determining if it has \((n-1)\) independent first integrals. We will introduce a new criterion for assessing the local and/or global integrability of real vector fields, of class \(C^r\), in finite dimension, and thereby we will show that there are extensive families of real vector fields that are \(C^r\) completely integrable, at a local or global level.

We study the geometry of positive cones of left-invariant total orders (left-order, for short) in finitely generated groups, focusing on connectivity properties. We will show that some hyperbolic group (e.g. free groups) do not support coarsely connected positive cones, and that this lack of connectivity is stable under some amalgamated free product.