Fixed Point of the Parabolic Renormalization Operator

  • Oscar E. Lanford III
  • Michael Yampolsky

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Oscar E. Lanford, Michael Yampolsky
    Pages 1-4
  3. Oscar E. Lanford, Michael Yampolsky
    Pages 5-44
  4. Oscar E. Lanford, Michael Yampolsky
    Pages 45-94
  5. Oscar E. Lanford, Michael Yampolsky
    Pages 95-104
  6. Oscar E. Lanford, Michael Yampolsky
    Pages 105-108
  7. Back Matter
    Pages 109-111

About this book


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 experts in the field as well as graduate students wishing to explore one of the frontiers of modern complex dynamics.


Ecalle-Voronin invariant Fatou coordinates Inou-Shishikura fixed point analytic map complex dynamics renormalization

Authors and affiliations

  • Oscar E. Lanford III
    • 1
  • Michael Yampolsky
    • 2
  1. 1.ZürichSwitzerland
  2. 2.University of Toronto, Department of MathematicsTorontoCanada

Bibliographic information

  • DOI
  • Copyright Information The Author(s) 2014
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-11706-5
  • Online ISBN 978-3-319-11707-2
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • About this book