Rotation Sets and Complex Dynamics

Benefits

• 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

Book

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

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

Introduction

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.

Keywords

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

• Book Title Rotation Sets and Complex Dynamics
• Authors Saeed Zakeri
• Series Title Lecture Notes in Mathematics
• Series Abbreviated Title Lect.Notes Mathematics
• DOI https://doi.org/10.1007/978-3-319-78810-4
• Copyright Information Springer International Publishing AG, part of Springer Nature 2018
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-3-319-78809-8
• eBook ISBN 978-3-319-78810-4
• Series ISSN 0075-8434
• Series E-ISSN 1617-9692
• Edition Number 1
• Number of Pages XIV, 124
• Number of Illustrations 2 b/w illustrations, 32 illustrations in colour
• Topics
• Buy this book on publisher's site
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