© 2018

Rotation Sets and Complex Dynamics


  • Provides the first systematic treatment of rotation sets

  • The abstract treatment is augmented by concrete examples of applications in polynomial dynamics

  • The clear and detailed exposition is accompanied by numerous illustrations, making it accessible to graduate students


Part of the Lecture Notes in Mathematics book series (LNM, volume 2214)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Saeed Zakeri
    Pages 1-15
  3. Saeed Zakeri
    Pages 17-33
  4. Saeed Zakeri
    Pages 35-56
  5. Saeed Zakeri
    Pages 57-83
  6. Saeed Zakeri
    Pages 85-118
  7. Back Matter
    Pages 119-124

About this book


This monograph examines rotation sets under the multiplication by d (mod 1) map and their relation to degree d polynomial maps of the complex plane. These sets are higher-degree analogs of the corresponding sets under the angle-doubling map of the circle, which played a key role in Douady and Hubbard's work on the quadratic family and the Mandelbrot set. Presenting the first systematic study of rotation sets, treating both rational and irrational cases in a unified fashion, the text includes several new results on their structure, their gap dynamics, maximal and minimal sets, rigidity, and continuous dependence on parameters. This abstract material is supplemented by concrete examples which explain how rotation sets arise in the dynamical plane of complex polynomial maps and how suitable parameter spaces of such polynomials provide a complete catalog of all such sets of a given degree. As a main illustration, the link between rotation sets of degree 3 and one-dimensional families of cubic polynomials with a persistent indifferent fixed point is outlined.

The monograph will benefit graduate students as well as researchers in the area of holomorphic dynamics and related fields.


Rotation Set Circle Map Rotation Number Rotation Cycle Complex Dynamics Julia Set Mandelbrot Set External Ray Doubling Map Dynamical Systems

Authors and affiliations

  1. 1.Department of MathematicsQueens College of CUNY, Queens, NY, USA; Department of Mathematics, Graduate Center of CUNYNew YorkUSA

Bibliographic information

Industry Sectors
Energy, Utilities & Environment
Finance, Business & Banking