Universal Mandelbrot Set, The: Beginning Of The Story

by Alexei Morozov, Valery Dolotin

Overview

This book is devoted to the structure of the Mandelbrot set — a remarkable and important feature of modern theoretical physics, related to chaos and fractals and simultaneously to analytical functions, Riemann surfaces, phase transitions and string theory. The Mandelbrot set is one of the bridges connecting the world of chaos and order.The authors restrict consideration to discrete dynamics of a single variable. This restriction preserves the most essential properties of the subject, but drastically simplifies computer simulations and the mathematical formalism.The coverage includes a basic description of the structure of the set of orbits and pre-orbits associated with any map of an analytic space into itself. A detailed study of the space of orbits (the algebraic Julia set) as a whole, together with related attributes, is provided. Also covered are: moduli space in the space of maps and the classification problem for analytic maps, the relation of the moduli space to the bifurcations (topology changes) of the set of orbits, a combinatorial description of the moduli space (Mandelbrot and secondary Mandelbrot sets) and the corresponding invariants (discriminants and resultants), and the construction of the universal discriminant of analytic functions in terms of series coefficients. The book concludes by solving the case of the quadratic map using the theory and methods discussed earlier.

Product Details

ISBN-13:	9789812568373
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/10/2006
Pages:	176
Product dimensions:	6.20(w) x 9.10(h) x 0.70(d)

Table of Contents

Preface     v
Introduction     1
Notions and notation     7
Objects, associated with the space X     7
Objects, associated with the space M     11
Combinatorial objects     16
Relations between the notions     19
Summary     21
Orbits and grand orbits     21
Mandelbrot sets     21
Forest structure     22
Relation to resultants and discriminants     24
Relation to stability domains     24
Critical points and locations of elementary domains     25
Perturbation theory and approximate self-similarity of Mandelbrot set     26
Trails in the forest     26
Sheaf of Julia sets over moduli space     27
Fragments of theory     31
Orbits and reduction theory of iterated maps     31
Bifurcations and discriminants: from real to complex     33
Discriminants and resultants for iterated maps     34
Period-doubling and beyond     36
Stability and Mandelbrot set     38
Towards the theory of Julia sets     39
Grand orbits and algebraic Julia sets     39
From algebraic to ordinary Julia set     40
Bifurcations of Julia set     41
On discriminant analysis for grand orbits     42
Decomposition formula for F[subscript n,s] (x; f)     42
Irreducible constituents of discriminants and resultants     43
Discriminant analysis at the level (n, s) = (1, 1): basic example     44
Sector (n, s) = (1, s)     46
Sector (n, s) = (2, s)     47
Summary     48
On interpretation of w[subscript n, k]     52
Combinatorics of discriminants and resultants     58
Shapes of Julia and Mandelbrot sets     60
Generalities     60
Exact statements about 1-parametric families of polynomials of power-d     62
Small-size approximation     63
Comments on the case of f[subscript c](x) = x[superscript d] + c     64
Analytic case     66
Discriminant variety D     69
Discriminants of polynomials     69
Discriminant variety in entire M     71
Discussion     72
Map f(x) = x[superscript 2] + c: from standard example to general conclusions     75
Map f(x) = x[superscript 2] + c. Roots and orbits, real and complex     76
Orbits of order one (fixed points)     76
Orbits of order two      77
Orbits of order three     79
Orbits of order four     80
Orbits of order five     82
Orbits of order six     83
Mandelbrot set for the family f[subscript c](x) = x[superscript 2] + c     85
Map f(x) = x[superscript 2] + c. Julia sets, stability and preorbits     87
Map f(x) = x[superscript 2] + c. Bifurcations of Julia set and Mandelbrot sets, primary and secondary     96
Conclusions about the structure of the "sheaf" of Julia sets over moduli space (of Julia sets and their dependence on the map f)     104
Other examples     111
Equivalent maps     112
Linear maps     112
The family of maps f[superscript Alpha Beta] = [Alpha] + [Beta]x     112
Multidimensional case     113
Quadratic maps     114
Diffeomorphic maps     114
Map f = x[superscript 2] + c     115
Map f [subscript gamma Beta 0] = [gamma]x[superscript 2] [Beta]x = [gamma]x[superscript 2] + (b + 1)x     117
Generic quadratic map and f = x[superscript 2] + px + q     119
Families as sections     120
Cubic maps     121
Map f[subscript p,q](x) = x[superscript 3] + px + q     123
Map f[subscript c] = x[superscript 3] + c      128
Map f[subscript c](x) = cx[superscript 3] + x[superscript 2]     132
f[subscript gamma] = x[superscript 3] + [gamma]x[superscript 2]     136
Map f[subscript a;c] = ax[superscript 3] + (1 - a)x[superscript 2] + c     139
Quartic maps     144
Map f[subscript c] = x[superscript 4] + c     144
Maps f[subscript d;c](x) = x[superscript d] + c     147
Generic maps of degree d, f(x) = [Characters not reproducible] [Alpha subscript i]x[superscript i]     150
Conclusion     157
Bibliography     161