Fractals and Chaos

by Mandelbrot, Benoit B.; Jones, P. W.; Evertsz, C. J. G. (CON); Gutzwiller, Martin C. (CON)

• ISBN13: 9780387201580
• ISBN10: 0387201580
• Format: Hardcover
• Copyright: 2004-01-01
• Publisher: Springer Verlag

Summary

This heavily illustrated book combines hard-to-find early papers by the author with additional chapters that describe the historical background and context. Key topics are quadratic dynamics and its Julia and Mandelbrot sets, nonquadratic dynamics, Kleinian limit sets, and the Minkowski measure. Book jacket.

Author Biography

Benoit B. Mandelbrot is Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) at the IBM T. J. Watson Research Center

Table of Contents

Foreword	ix
Peter W. Jones
Preface	1	(8)
I QUADRATIC JULIA AND MANDELBROT SETS
Introduction to papers on quadratic dynamics: a progression from seeing to discovering (2003)	9	(18)
Acknowledgments related to quadratic dynamics (2003)	27	(10)
Fractal aspects of the iteration of z → λ z (1-z) for complex λ and z (M1980n)	37	(15)
Cantor and Fatou dusts ; self-squared dragons (M 1982F)	52	(21)
The complex quadratic map and its M-set (M1983p)	73	(23)
Bifurcation points and the ``n squared'' approximation and conjecture (M1985g)	96	(4)
M.L. Frame
K. Mitchell
The ``normalized radical'' of the M-set (M1985g)	100	(10)
The boundary of the M-set is of dimension 2 (M1985g)	110	(4)
Certain Julia sets include smooth components (M1985g)	114	(3)
Domain-filling sequences of Julia sets, and intuitive rationale for the Siegel discs (M1985g)	117	(8)
Continuous interpolation of the quadratic map and intrinsic tiling of the interiors of Julia sets (M1985n)	125	(12)
II NONQUADRATIC RATIONAL DYNAMICS
Introduction to chaos in nonquadratic dynamics: rational functions devised from doubling formulas (2003)	137	(9)
The map z → λ (z + 1/ z) and roughening of chaos from linear to planar (computer-assisted homage to K. Hokusai) (M1984k)	146	(11)
Two nonquadratic rational maps, devised from Weierstrass doubling formulas (1979-2003)	157	(14)
III ITERATED NONLINEAR FUNCTION SYSTEMS AND THE FRACTAL LIMIT SETS OF KLEINIAN GROUPS
Introduction to papers on Kleinian groups, their fractal limit sets, and IFS: history, recollections, and acknowledgments (2003)	171	(7)
Self-inverse fractals, Apollonian nets, and soap (M 1982F)	178	(15)
Symmetry by dilation or reduction, fractals, roughness (M2002w)	193	(12)
Self-inverse fractals osculated by sigma-discs and limit sets of inversion (``Kleinian'') groups (M1983m)	205	(16)
IV MULTIFRACTAL INVARIANT MEASURES
Introduction to measures that vanish exponentially almost everywhere: DLA and Minkowski (2003)	221	(10)
Invariant multifractal measures in chaotic Hamiltonian systems and related structures (Gutzwiller & M 1988)	231	(8)
The Minkowski measure and multifractal anomalies in invariant measures of parabolic dynamic systems (M1993s)	239	(12)
Harmonic measure on DLA and extended self-similarity (M & Evertsz 1991)	251	(8)
V BACKGROUND AND HISTORY
The inexhaustible function z squared plus c (1982-2003)	259	(9)
The Fatou and Julia stories (2003)	268	(8)
Mathematical analysis while in the wilderness (2003)	276	(5)
Cumulative Bibliography	281	(18)
Index	299