under construction
Intro mention that it is a spiritual sequel for the Romance in the upper half plane article. There we analysed hyperbolic geometry x modular forms. Here we do hyperbolic geometry x complex dynamics. The title might be misleading as we in fact analyse using the disc more in this article, unlike the upper half plane in the previous one.
First start with the review of Complex Analysis - focus on Schwarz lemma (and Schwarz Pick lemma) - its importance in classifying the automorphisms of $\mathbb{D}$ and also in analysing the Poincare (hyperbolic) metric. Importance of Poincare metric - preserving lengths of conformal maps that we get using Schwarz lemma, Riemann mapping and uniformization theorems. Why Poincare metric is needed? Why the Euclidean metric won't suffice?
Rational maps, give visualization using $z \mapsto z^2$ example , Conjugacy, enough to study rational maps on the disc (thanks to Riemann mapping), critical points and fixed points, multipliers and why the biholomorphic bijection condition is needed? why holomorphy is needed?, Denjoy Wolff theorem, fixed points of rational maps and hyperbolic geometry, Denjoy Wolff for multiple connected domains.
Tell about further reading - Fatou, Julia sets etc and further connections like Kleinian groups
Mention about the works of Sullivan - series of 3 papers. The first paper speaks of the analogy between perturbation of complex parameter (starting with c=0) of the quadratic map and how Julia sets changes accordingly and the theory of Poincare regarding how a small perturbation of the Fuchsian group (starting with PSL_{2}(Z)) reflects accordingly in the change of the limit set of the corresponding Fuchsian group. In fact in some sense, the Julia sets of rational maps have some analogy with the limit sets of Fuchsian groups in terms of properties. Sullivan dictionary - open article with this in the intro - groundbreaking three papers of Sullivan, contributions of Poincare, Fatou and Julia in this regard as the history part. These results actually work on the hyperbolic 3 space and not the usual upper half plane. There are Fuchsian groups and Kleinian groups associated to this space. Check relevant materials.
Refer the book "Holomorphic dynamics" by Morosawa, et al for content reg things mentioned in first and second para above
