Geometry of the feigenbaum map

Abstract

We show that the Cvitanović-Feigenbaum equation can be inter-preted as a linearizing equation, and the domain of analyticity of the Feigen-baum fixed point of renormalization as a basin of attraction. As a consequence, we give a combinatorial description of this ramified covering, and we show the surprising result that there exist points in the boundary of this domain with three accesses from inside the domain. Besides, there is a natural decomposi- tion of this basin which makes it possible to recover a result of local connec-tivity by Hu and Jiang (The Julia set of the Feigenbaum quadratic polynomial is locally connected, Preprint, 1993) for the Feigenbaum Julia set. © 1999 American Mathematical Society.