According to Royden-Gardiner theorem, any holomorphic mapping between Teichmuller spaces does not expand the Teichmuller distance. (Just like Schwarz-Pick theorem says that any holomorphic mapping between hyperbolic Riemann surfaces does not expand the Poincare metric.) In the theory of renormalizations, one often wants to show that the renormalization map (which is defined in a transcendental way) on the space of certain dynamical systems is hyperbolic or contracting.

Therefore Royden-Gardiner theorem is an obvious candidate of tools to obtain the contraction. This idea was first used by Sullivan in his work on generalized Feigenbaum type renormalization. In this talk, we will discuss two applications of Teichmuller theory to renormalizations:

1. Parabolic renormalization for parabolic fixed points and their perturbations. 2. Rigidity of real quadratic polynomials only using Yoccoz's combinatorial a priori bounds.

For these cases, we deal with the Teichmuller spaces of a disk or a punctured disk and show that a certain inclusion map induces a contraction in Teichmuller distance.