Renormalization Ideas in Conformal Dynamics

1. Introduction How to look at a dynamical system f at a small scale? You should take a small piece of the phase space, consider the rst return map to this piece, and then rescale it to \the original size". The new dynamical system is called the renormalization Rf of the original one. It may happen that Rf looks \similar" to f, and then you can try to repeat this procedure, and construct the second renormalization R 2 f, etc. Asymptotic properties of this sequence of renormalizations reeect micro-structure of the original system. For example, convergence of the sequence R n f to a map f independent of f (from some class of similar maps) means that all maps of this class have in small scales a universal geometry represented by f. A striking phenomenon of this kind is the Feigenbaum-Coullet-Tresser Universality Law ((CT, F], see McM1], x6). It deals with the class of suuciently smooth unimodal maps of an interval I with the critical point 0 of a given type jxj d (\unimodal" means: \with one critical point"). Under some combinatorial assumptions on the positions of the rst four iterates of the critical point, the interval J = ?f 2 0; f 2 0] turns out to be invariant under f 2. Moreover f 2 jJ is again a unimodal map of the same class. Rescaling J to the original size, we obtain the \doubling renormalization" Rf of f. A map f of such kind can be called \renormalizable". If it happens that this procedure can be repeated, we have twice renormalizable maps, etc. The Universality Law asserts that the renormalizations R n f of innnitely renormalizable maps converge to a map f independent of f. Thus all innnitely renormalizable unimodal maps with a given type of the critical point have asymptotically the same geometry in small scales. A similar picture is observed not only for the doubling renormalization but for other periods as well. We have here a kind of the rigidity phenomenon: Combinatorics of an object determines its geometry. Compare it with the Rigidity Conjecture discussed by McMullen McM1]. The latter is concerned with a nitely dimensional family of globally deened objects, rational maps. The rigidity conclusion is also global: the geometry of the whole Julia set is determined by combinatorics. In the Feigenbaum-Coullet-Tresser