Thurston's theorem states that a post-critically finite branched self-covering of a sphere is combinatorially equivalent to a rational map if and only if it does not admit a certain kind of topological obstruction. The proof of the theorem uses iteration on the Teichmueller space, the so-called Thurston pullback map, to find an invariant complex structure. In the obstructed (and non-parabolic) case, the map diverges to the stratum at infinity associated with the canonical obstruction of the map. We will review properties of the Thurston pullback map and discuss its relation to the twisting problem of branched self-coverings of the sphere and other applications.

In this introductory course, we will present the classical theory of local complex dynamics in one and several complex variables. The first two lectures will cover background material and classical results. The last two lectures aim to explore more recent developments in higher dimensions with some applications, such as the existence of wandering Fatou components and parabolic curves.

Thurston's theorem states that a post-critically finite branched self-covering of a sphere is combinatorially equivalent to a rational map if and only if it does not admit a certain kind of topological obstruction. The proof of the theorem uses iteration on the Teichmueller space, the so-called Thurston pullback map, to find an invariant complex structure. In the obstructed (and non-parabolic) case, the map diverges to the stratum at infinity associated with the canonical obstruction of the map. We will review properties of the Thurston pullback map and discuss its relation to the twisting problem of branched self-coverings of the sphere and other applications.

In this introductory course, we will present the classical theory of local complex dynamics in one and several complex variables. The first two lectures will cover background material and classical results. The last two lectures aim to explore more recent developments in higher dimensions with some applications, such as the existence of wandering Fatou components and parabolic curves.

"A priori bounds" provide us with a uniform geometric control of dynamical systems in all scales. Central problems of Holomorphic Dynamics, like universal self-similarity of dynamical and parameter pictures, MLC, topological structure of neutral maps, and the area problem for Julia sets, depend on such bounds. We will give an overview of advances in this problem over the past 30 years.

Approximation Theory has been one of the most important tools for constructing functions in Complex Dynamics. In this mini-course, we will give an overview of results and techniques from Approximation Theory that have been used in the area, including Runge’s theorem and Arakelyan's theorem. We will present several versions of these results which help us construct transcendental entire and meromorphic functions with specific properties. We will also give an overview of a paper by Eremenko and Lyubich, which includes some of the earliest applications of Approximation Theory in Transcendental Dynamics.

In the past few years, there has been a resurge in the use of Approximation Theory to obtain wandering domains with interesting dynamics and topology. Benini, Evdoridou, Fagella, Rippon and Stallard constructed examples of wandering domains with several types of internal dynamics that were unknown before. More recently, Marti-Pete, Rempe and Waterman, inspired by work of Boc Thaler, constructed wandering domains with interesting topology, including wandering domains that form Lakes of Wada. We will discuss these two constructions in detail and present some open questions in this area.

The dynamics inside periodic components of the stable set has a strong link with classical theorems of complex analysis like the Denjoy-Wolff Theorem about analytic maps of the unit disk. The fractal boundaries of such components arising so naturally from iteration often present interesting topological properties which may play a role when trying to transfer results from the unit disk back to the dynamical plane. However, if the components are not periodic but wandering, we need to reach further and consider non-autonomous iteration. Starting from periodic components, I aim to present some recent results about the dynamics inside wandering domains and also on their boundaries. Many of the results are proven in the very general setting of non-autonomous dynamics or even for sequences of holomorphic maps.

Approximation Theory has been one of the most important tools for constructing functions in Complex Dynamics. In this mini-course, we will give an overview of results and techniques from Approximation Theory that have been used in the area, including Runge’s theorem and Arakelyan's theorem. We will present several versions of these results which help us construct transcendental entire and meromorphic functions with specific properties. We will also give an overview of a paper by Eremenko and Lyubich, which includes some of the earliest applications of Approximation Theory in Transcendental Dynamics.

In the past few years, there has been a resurge in the use of Approximation Theory to obtain wandering domains with interesting dynamics and topology. Benini, Evdoridou, Fagella, Rippon and Stallard constructed examples of wandering domains with several types of internal dynamics that were unknown before. More recently, Marti-Pete, Rempe and Waterman, inspired by work of Boc Thaler, constructed wandering domains with interesting topology, including wandering domains that form Lakes of Wada. We will discuss these two constructions in detail and present some open questions in this area.