Fractals and Chaos: The Mandelbrot Set and BeyondIt has only been a couple of decades since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot set. That picture, now seeming graphically primitive, has changed our view of the mathematical and physical universe. The properties and circumstances of the discovery of the Mandelbrot Set continue to generate much interest in the research community and beyond. This book contains the hard-to-obtain original papers, many unpublished illustrations dating back to 1979 and extensive documented historical context showing how Mandelbrot helped change our way of looking at the world. |
Contents
QUADRATIC JULIA AND MANDELBROT SETS | 9 |
C2 Acknowledgments related to quadratic dynamics 2003 | 27 |
C3 Fractal aspects of the iteration of z λ z 1z | 37 |
C4 Cantor and Fatou dusts selfsquared dragons M 1982F | 52 |
C5 The complex quadratic map and its Mset M1983p | 73 |
C6 Bifurcation points and the n squared approximation | 96 |
C8 The boundary of the Mset is of dimension 2 M1985g | 110 |
C10 Domainfilling sequences of Julia sets | 117 |
C14 Two nonquadratic rational maps devised | 157 |
ITERATED NONLINEAR FUNCTION SYSTEMS | 171 |
C17 Symmetry by dilation or reduction fractals roughness M2002w | 193 |
MULTIFRACTAL INVARIANT MEASURES | 221 |
C21 The Minkowski measure and multifractal anomalies | 239 |
BACKGROUND AND HISTORY | 259 |
C25 Mathematical analysis while in the wilderness 2003 | 276 |
| 299 | |
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Common terms and phrases
algorithm Apollonian approximation atom attractor bifurcation boundary Bourbaki C₁ C₂ called Cantor dust chaos chaotic Chapter foreword circle cluster complex numbers conjecture construction continuous converge corresponding defined denoted described distribution domain Douady dragon dynamical system example Fatou Fatou-Julia Figure finite fixed point fractal curve fractal dimension fractal geometry Fuchsian function Gaston Julia Gutzwiller harmonic measure hence Hölder illustrations infinite infinity intersect interval intrinsic tiling invariant inverse island molecules iteration J-set Julia set Klein Kleinian groups Lattès limit cycle limit points limit set linear M-set Mandelbrot set mathematicians mathematics Minkowski measure multifractal multifractal measures Myrberg notion observations orbit osculating paper parameter Peano physics Pierre Fatou Poincaré properties quadratic map radius random rational real axis roughness self-inverse set self-similarity self-squared sequence shape Siegel disc structure subradical symmetry tangent theory topic transform values yield