The study of iterations of rational mappings of the Riemann sphere was started at the end of 19-th - beginning of 20th century, in the works of Fatou and Julia. The Riemann sphere is split into two invariant sets:
1) the Fatou set, which is an open set of points where the iterations are Lyapunov stable;
2) the Julia set, which is its complement and is the closure of the set of repelling periodic points (see [L]). The classical theorem of D.Sullivan [Su] says that each connected component of the Fatou set always becomes periodic after some iteration.
A rational mapping is said to be hyperbolic, if the positive orbit of each critical point converges to an attracting cycle of periodic points.
R.Mane, P.Sad and D.Sullivan have proved in their joint paper [M] that each hyperbolic mapping is structurally stable on its Julia set: the mapping and its small perturbation (in the class of rational mappings of the same degree) are conjugated on their Julia sets by a homeomorphism}. The key-lemma of their proof has generated the theory of holomorphic motions and have inspired the principal result of Z.Slodkowski [S] in this theory saying that any holomorphic motion of a subset of the Riemann sphere can be extended up to the global holomorphic motion of the whole sphere. This means that any (not necessarily finite) collection $\{(f_i(z),z)\}\subset\overline{\Bbb C}\times D$, $D=\{|t|<1\}$, of disjoint graphs of holomorphic functions $f_i:D\to\overline{\Bbb C}$ extends up to filling in of the whole product $\overline{\Bbb C}\times D$ by disjoint graphs of holomorphic functions.
Holomorphic motions are related to the Teichmueller theory: the geometry of the space of Riemann surfaces (say, of fixed genus). The space of Riemann surfaces is a complex-analytic manifold. The Teichmueller distance between Riemann surfaces induces a Finsler metric on this manifold. On the other hand, the manifold carries the Kobayashi metric, which is invariant under its biholomorphic automorphisms. Royden's theorem [Ga] says that these two metrics coincide. This yields the classification of the automorphisms of the space of Riemann surfaces.
In the minicours we are going to talk about the results mentioned above and we'll discuss the relation of holomorphic motions and Royden's theorem.
[Ga] Gardiner, F. Teichmueller theory and quadratic differentials. - John Wiley and Sons, 1987.
[L] Lyubich, M. Yu. Dynamics of rational transformations: topological picture. - Russian Math. Surveys 41 (1986), no. 4, 43--117.
[M] MaЯИ, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps. - Ann. Sci. иcole Norm. Sup. (4) 16 (1983), no. 2, 193--217.
[S] Slodkowski, Z. Extensions of holomorphic motions. - Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 2, 185--210.
[Su] Sullivan, D. Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains. - Ann. of Math. (2) 122 (1985), no. 3, 401--418.
