Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set
Keller, Karsten.
Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set [electronic resource] / by Karsten Keller. - XII, 208 p. online resource. - Lecture Notes in Mathematics, 1732 0075-8434 ; . - Lecture Notes in Mathematics, 1732 .
1. Introduction: Quadratic iteration and Julia equivalences. The Mandelbrot set -- 2. Abstract Julia sets: Symbolic dynamics of the angle-doubling map. Invariant laminations. Julia equivalences -- 3. The Abstract Mandelbrot set: The Abstract Mandelbrot set - an atlas of Abstract Julia sets. The ordered Abstract Mandelbrot set. Renormalization. Correspondence and Translation Principles -- 4. Abstract and concrete theory: Quadratic iteration. Miscellaneous. Appendix: Invariant and completely invariant factors. Simple statements. Shift-invariant factors. Further interesting examples.
This book is mainly devoted to the combinatorics of quadratic holomorphic dynamics. The conceptual kernel is a self-contained abstract counterpart of connected quadratic Julia sets which is built on Thurston's concept of a quadratic invariant lamination and on symbolic descriptions of the angle-doubling map. The theory obtained is illustrated in the complex plane. It is used to give rigorous proofs of some well-known and some partially new statements on the structure of the Mandelbrot set. The text is intended for graduate students and researchers. Some elementary knowledge in topology and in functions of one complex variable is assumed.
9783540455899
10.1007/BFb0103999 doi
Differential equations, partial.
Topology.
Partial Differential Equations.
Topology.
QA370-380
515.353
Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set [electronic resource] / by Karsten Keller. - XII, 208 p. online resource. - Lecture Notes in Mathematics, 1732 0075-8434 ; . - Lecture Notes in Mathematics, 1732 .
1. Introduction: Quadratic iteration and Julia equivalences. The Mandelbrot set -- 2. Abstract Julia sets: Symbolic dynamics of the angle-doubling map. Invariant laminations. Julia equivalences -- 3. The Abstract Mandelbrot set: The Abstract Mandelbrot set - an atlas of Abstract Julia sets. The ordered Abstract Mandelbrot set. Renormalization. Correspondence and Translation Principles -- 4. Abstract and concrete theory: Quadratic iteration. Miscellaneous. Appendix: Invariant and completely invariant factors. Simple statements. Shift-invariant factors. Further interesting examples.
This book is mainly devoted to the combinatorics of quadratic holomorphic dynamics. The conceptual kernel is a self-contained abstract counterpart of connected quadratic Julia sets which is built on Thurston's concept of a quadratic invariant lamination and on symbolic descriptions of the angle-doubling map. The theory obtained is illustrated in the complex plane. It is used to give rigorous proofs of some well-known and some partially new statements on the structure of the Mandelbrot set. The text is intended for graduate students and researchers. Some elementary knowledge in topology and in functions of one complex variable is assumed.
9783540455899
10.1007/BFb0103999 doi
Differential equations, partial.
Topology.
Partial Differential Equations.
Topology.
QA370-380
515.353