The Mandelbrot Set, Theme and Variations
The Mandelbrot set is a fractal shape that classifies the dynamics of quadratic polynomials. It has a remarkably rich geometric and combinatorial structure. This volume provides a systematic exposition of current knowledge about the Mandelbrot set and presents the latest research in complex dynamics. Topics discussed include the universality and the local connectivity of the Mandelbrot set, parabolic bifurcations, critical circle homeomorphisms, absolutely continuous invariant measures and matings of polynomials, along with the geometry, dimension and local connectivity of Julia sets. In addition to presenting new work, this collection documents important results hitherto unpublished or difficult to find in the literature. This book will be of interest to graduate students in mathematics, physics and mathematical biology, as well as researchers in dynamical systems and Kleinian groups.
- Documents important results about the Mandelbrot set and related topics which were hitherto unpublished or difficult to find in the current literature
- Covers both new research and the basic knowledge of the subject
- Unique treatment of the Mandelbrot set, Julia sets and dynamical systems
Reviews & endorsements
'… this collection presents important results hitherto unpublished or difficult to find in the literature.' European Maths Society Journal
'The analytic techniques employed cover an exceptionally broad range and students of mainstream science in search of an appropriate mathematical model to fit their dynamical scheme will find a very solid theoretical base going way beyond the basic concepts.' Contemporary Physics
… should be studied in depth by any potential worker in this field. This book should remain popular for many years to come.' The Mathematical Gazette
Product details
April 2000Paperback
9780521774765
388 pages
230 × 153 × 22 mm
0.525kg
61 b/w illus.
Available
Table of Contents
- Introduction L.Tan
- Preface J. Hubbard
- 1. The Mandelbrot set is universal C. McMullen
- 2. Baby Mandelbrot sets are born in cauliflowers A. Douady, X. Buff, R. Devaney and P. Sentenac
- 3. Modulation dans l'ensemble de Mandelbrot P. Haïssinsky
- 4. Local connectivity of Julia sets: expository lectures J. Milnor
- 5. Holomorphic motions and puzzles (following M. Shishikura) P. Roesch
- 6. Local properties of the Mandelbrot set at parabolic points L.Tan
- 7. Convergence of rational rays in parameter spaces C. Petersen and G. Ryd
- 8. Bounded recurrence of critical points and Jakobson's Theorem S. Luzzatto
- 9. The Herman–Swiatek theorems with applications C. Petersen
- 10. Perturbations d'une fonction linéarisable H. Jellouli
- 11. Indice holomorphe et multiplicateur H. Jellouli
- 12. An alternative proof of Mañé's theorem on non-expanding Julia sets M. Shishikura and L.Tan
- 13. Geometry and dimension of Julia sets Y. -C. Yin
- 14. On a theorem of Mary Rees for the matings of polynomials M. Shishikura
- 15. Le théorème d'intégrabilité des structures presque complexes A. Douady and X. Buff
- 16. Bifurcation of parabolic fixed points M. Shishikura.
