Fixed Point of the Parabolic Renormalization Operator

This monograph grew out of the authors' efforts to provide a natural geometric description for the class of maps invariant under parabolic renormalization and for the Inou-Shishikura fixed point itself as well as to carry out a computer-assisted study of the parabolic renormalization operator. It introduces a renormalization-invariant class of analytic maps with a maximal domain of analyticity and rigid covering properties and presents a numerical scheme for computing parabolic renormalization of a germ, which is used to compute the Inou-Shishikura renormalization fixed point.

Inside, readers will find a detailed introduction into the theory of parabolic bifurcation, Fatou coordinates, Écalle-Voronin conjugacy invariants of parabolic germs, and the definition and basic properties of parabolic renormalization.

The systematic view of parabolic renormalization developed in the book and the numerical approach to its study will be interesting to both expertsin the field as well as graduate students wishing to explore one of the frontiers of modern complex dynamics.

The first detailed introduction into one of the cutting-edge subjects of modern Complex Dynamics A new numerical approach to computing Fatou coordinates of a parabolic germ Text illustrated with many detailed computer-generated images Includes supplementary material: sn.pub/extras

Autorentext
Michael Yampolsky is an expert in Dynamical Systems, particularly in Holomorphic Dynamics and Renormalization Theory.

Inhalt
1 Introduction.- 2 Local dynamics of a parabolic germ.- 3 Global theory.- 4 Numerical results.- 5 For dessert: several amusing examples.- Index.