A major goal in complex dynamics is to understand dynamical moduli spaces; that is, conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to the complex plane. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. Many tools from complex analysis that pave the way for key breakthroughs in the one-dimensional setting do not carry over to higher dimensions. So instead of considering the whole moduli space, we follow an approach initiated by William Thurston and investigate special subvarieties of moduli space that give rise to dynamical moduli spaces. In this talk, we will explore the topology and geometry of the dynamical moduli spaces that play a prominent role in complex dynamics.

We study cubic polynomial maps from $\C$ to $\C$ with a critical orbit of period $p$. For each $p>0$ the space of conjugacy classes of such maps forms a smooth Riemann surface with a smooth compactification $\overline S_p$. For each $q>0$ I will describe a dynamically defined tessellation of $\overline S_p$. Each face of this tessellation corresponds to one particular behavior for periodic orbits of period $q$. (Joint work with John Milnor.)

Thurston proved that a non-Lattés branched cover of the sphere to itself is either equivalent to a rational map (that is: conjugate via a mapping class), or has a topological obstruction. The Nielsen–Thurston classification of mapping classes is an analogous theorem in low-dimensional topology. We unify these two theorems with a single proof, further connecting techniques from surface topology and complex dynamics. This is joint work with Jim Belk and Dan Margalit.

The moduli space M_{0,n} parametrizes point-configurations on the Riemann sphere. Hurwitz spaces parametrize branched covers between Riemann Surfaces, with prescribed ramification. I will describe ways in which M_{0,n} and Hurwitz spaces arise naturally when studying post-critically finite and “almost post-critically finite” rational maps.

Bifurcations arise when there is a drastic change in the solutions of some equation depending on a parameter, as the parameter varies. In this talk we study bifurcations in holomorphic families of meromorphic maps with finitely many singular values. The equation(s) that we will study are the equations defining periodic points of period n. Such equations are crucial in complex dynamics because the Julia set (the set on which the dynamics is chaotic) is the closure of repelling periodic points. The celebrated results by Mane-Sad-Sullivan for families of rational maps (and independently by Lyubich, and by Levin for polynomials) show that in a set of parameters where no bifurcations of periodic points occur, the Julia set stays almost the same and so does the dynamics; precisely speaking, all maps are topologically conjugate in such set. Moreover, they establish a precise correlation between bifurcations of periodic points and a change of behaviour in the orbits of singular values. The key new feature that appears for families of meromorphic maps is that periodic points can disappear at infinity at specific parameters, creating a new type of bifurcations. Our work connects this new type of bifurcations with change of behaviour in singular orbits, to establish Mane-Sad-Sullivan's Theorem for meromorphic maps. This is joint work with Matthieu Astorg and Nùria Fagella.

In the study of the dynamics of a transcendental entire function f, we aim to describe its locus of chaotic behaviour, known as its Julia set and denoted by J(f). For many such f, the Julia set is a collection of unbounded curves that escape to infinity under iteration and form a Cantor bouquet, i.e., a subset of the complex plane ambiently homeomorphic to a straight brush. We show that there exists f whose Julia set J(f) is a collection of escaping curves, but J(f) is not a Cantor bouquet. On the other hand, we prove for certain f that if J(f) contains an absorbing Cantor bouquet, that is, a Cantor bouquet to which all escaping points are eventually mapped, then J(f) must be a Cantor bouquet. This is joint work with L. Rempe.

Let U be a Fatou component of a transcendental entire function. If U is not eventually periodic then it is called a wandering domain. Although Sullivan's celebrated result showed that rational maps have no wandering domains, transcendental entire functions can have wandering domains. In this talk we will look at internal dynamics of simply connected wandering domains and we will give a nine-way classification of these domains. We will also show that all nine types of wandering domains that arise from this classification are realizable. This is joint work with Anna Miriam Benini, Nuria Fagella, Phil Rippon and Gwyneth Stallard.

A hyperbolic rational map in one complex variable provides a natural uniformly hyperbolic dynamical system if we restrict the map to its Julia set. In this talk, I will explain how ergodic theoretic methods can help us understand (hyperbolic) rational maps and (hyperbolic components of) the moduli space. This is based on joint work with Hongming Nie.