The universal mandelbrot set : beginning of the story

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.Customer ReviewsAverage Rating: Customer Rating for this product is out of 5Be the first to write a review!From the PublisherThis 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: 9812568379 \* ISBN-13: 9789812568373 \* Format: Hardcover, 162pp \* Publisher: World Scientific Publishing Company, Incorporated \* Pub. Date: October 2006 \* Sales Rank: 585,819 \* Table of ContentsTable of ContentsPreface vIntroduction 1Notions and notation 7Objects, associated with the space X 7Objects, associated with the space M 11Combinatorial objects 16Relations between the notions 19Summary 21Orbits and grand orbits 21Mandelbrot sets 21Forest structure 22Relation to resultants and discriminants 24Relation to stability domains 24Critical points and locations of elementary domains 25Perturbation theory and approximate self-similarity of Mandelbrot set 26Trails in the forest 26Sheaf of Julia sets over moduli space 27Fragments of theory 31Orbits and reduction theory of iterated maps 31Bifurcations and discriminants: from real to complex 33Discriminants and resultants for iterated maps 34Period-doubling and beyond 36Stability and Mandelbrot set 38Towards the theory of Julia sets 39Grand orbits and algebraic Julia sets 39From algebraic to ordinary Julia set 40Bifurcations of Julia set 41On discriminant analysis for grand orbits 42Decomposition formula for F[subscript n,s] (x; f) 42Irreducible constituents of discriminants and resultants 43Discriminant analysis at the level (n, s) = (1, 1): basic example 44Sector (n, s) = (1, s) 46Sector (n, s) = (2, s) 47Summary 48On interpretation of w[subscript n, k] 52Combinatorics of discriminants and resultants 58Shapes of Julia and Mandelbrot sets 60Generalities 60Exact statements about 1-parametric families of polynomials of power-d 62Small-size approximation 63Comments on the case of f[subscript c](x) = x[superscript d] + c 64Analytic case 66Discriminant variety D 69Discriminants of polynomials 69Discriminant variety in entire M 71Discussion 72Map f(x) = x[superscript 2] + c: from standard example to general conclusions 75Map f(x) = x[superscript 2] + c. Roots and orbits, real and complex 76Orbits of order one (fixed points) 76Orbits of order two 77Orbits of order three 79Orbits of order four 80Orbits of order five 82Orbits of order six 83Mandelbrot set for the family f[subscript c](x) = x[superscript 2] + c 85Map f(x) = x[superscript 2] + c. Julia sets, stability and preorbits 87Map f(x) = x[superscript 2] + c. Bifurcations of Julia set and Mandelbrot sets, primary and secondary 96Conclusions about the structure of the "sheaf" of Julia sets over moduli space (of Julia sets and their dependence on the map f) 104Other examples 111Equivalent maps 112Linear maps 112The family of maps f[superscript Alpha Beta] = [Alpha] + [Beta]x 112Multidimensional case 113Quadratic maps 114Diffeomorphic maps 114Map f = x[superscript 2] + c 115Map f [subscript gamma Beta 0] = [gamma]x[superscript 2] [Beta]x = [gamma]x[superscript 2] + (b + 1)x 117Generic quadratic map and f = x[superscript 2] + px + q 119Families as sections 120Cubic maps 121Map f[subscript p,q](x) = x[superscript 3] + px + q 123Map f[subscript c] = x[superscript 3] + c 128Map f[subscript c](x) = cx[superscript 3] + x[superscript 2] 132f[subscript gamma] = x[superscript 3] + [gamma]x[superscript 2] 136Map f[subscript a;c] = ax[superscript 3] + (1 - a)x[superscript 2] + c 139Quartic maps 144Map f[subscript c] = x[superscript 4] + c 144Maps f[subscript d;c](x) = x[superscript d] + c 147Generic maps of degree d, f(x) = [Characters not reproducible] [Alpha subscript i]x[superscript i] 150Conclusion 157Bibliography 161Customers who bought this also bought \* How Much Is a Million?Steven Kellogg, David M. Schwartz, Steven Kellogg \* Zero : The Biography of a Dangerous IdeaCharles Seife, Matt Zimet \* The Joy of PiDavid Blatner \* Number Theory for ComputingS. Y. Y. Yan, Martin E. Hellman, M. E. Hellmann \* Number Theory and Its HistoryOystein Ore, Ystein OreMore by This Authorloading animation loading... Direct Look At The Celebrated Chaotic Mandelbrot Set (in Fig. 1) Immediately Reveals That It Is A Collection Of Almost Ideal Circles And Cardioids, Unified In A Specific Forest Structure. In The Paper Arxiv:hep-th/0501235, A Systematic Algebro-geometric Approach Was Developed To The Study Of Generic Mandelbrot Sets, But Emergency Of Nearly Ideal Circles In The Special Case Of The Family X2 + C Was Not Fully Explained. In The Present, Paper, The Shape Of The Elementary Constituents Of Mandelbrot Set Is Explicitly Calculated, And Difference Between The Shapes Of Root And Descendant Domains (cardioids And Circles Respectively) Is Explained. Such Qualitative Difference Persists For All Other Mandelbrot Sets: Descendant Domains Always Have One Less Cusp Than The Root Ones. Details Of The Phase Transition Between Different Mandelbrot Sets Are Explicitly Demonstrated, Including Overlaps Between Elementary Domains And Dynamics Of Attraction/repulsion Regions. Explicit Examples Of Three-dimensional Sections Of Universal Mandelbrot Set Are Given. Also A Systematic Small-size Approximation Is Developed For Evaluation Of Various Feigenbaum Indices.--publisher's Website. 1. Introduction -- 2. Notions And Notation -- 3. Summary -- 4. Fragments Of Theory -- 5. Map F(x)=x[superscript 2] + C : From Standard Example To General Conclusions -- 6. Other Examples -- 7. Conclusion. V. Dolotin, A. Morozov. Includes Bibliographical References (p. 161-162). 5.1.2 Orbits of order two5.1.3 Orbits of order three; 5.1.4 Orbits of order four; 5.1.5 Orbits of order five; 5.1.6 Orbits of order six; 5.2 Mandelbrot set for the family fc(x) = x2 + c; 5.3 Map f(x) = x2 + c. Julia sets stability and preorbits; 5.4 Map f(x) = x2 + c. Bifurcations of Julia set and Mandelbrot sets primary and secondary; 5.5 Conclusions about the structure of the ""sheaf"" of Julia sets over moduli space (of Julia sets and their dependence on the map f); 6. Other examples; 6.1 Equivalent maps; 6.2 Linear maps; 6.2.1 The family of maps faB = a + Bx; 6.2.2 Multidimensional case Preface; Contents; 1. Introduction; 2. Notions and notation; 2.1 Objects associated with the space X; 2.2 Objects associated with the space M; 2.3 Combinatorial objects; 2.4 Relations between the notions; 3. Summary; 3.1 Orbits and grand orbits; 3.2 Mandelbrot sets; 3.2.1 Forest structure; 3.2.2 Relation to resultants and discriminants; 3.2.3 Relation to stability domains; 3.2.4 Critical points and locations of elementary domains; 3.2.5 Perturbation theory and approximate self-similarity of Mandelbrot set; 3.2.6 Trails in the forest; 3.3 Sheaf of Julia sets over moduli space 4.8 Combinatorics of discriminants and resultants4.9 Shapes of Julia and Mandelbrot sets; 4.9.1 Generalities; 4.9.2 Exact statements about 1-parametric families of polynomials of power-d; 4.9.3 Small-size approximation; 4.9.4 Comments on the case of fc(x) = xd + c; 4.10 Analytic case; 4.11 Discriminant variety D; 4.11.1 Discriminants of polynomials; 4.11.2 Discriminant variety in entire M; 4.12 Discussion; 5. Map f(x) = x2 + c: from standard example to general conclusions; 5.1 Map f(x) = x2 + c. Roots and orbits real and complex; 5.1.1 Orbits of order one (fixed points) 4. Fragments of theory4.1 Orbits and reduction theory of iterated maps; 4.2 Bifurcations and discriminants: from real to complex; 4.3 Discriminants and resultants for iterated maps; 4.4 Period-doubling and beyond; 4.5 Stability and Mandelbrot set; 4.6 Towards the theory of Julia sets; 4.6.1 Grand orbits and algebraic Julia sets; 4.6.2 From algebraic to ordinary Julia set; 4.6.3 Bifurcations of Julia set; 4.7 On discriminant analysis for grand orbits; 4.7.2 Irreducible constituents of discriminants and resultants; 4.7.6 Summary; 4.7.7 On interpretation of wntk

This book provides an overview of recent developments in web theory. Webs (i.e. families of foliations in general position) appear in many different fields of mathematics (differential geometry, algebraic geometry, differential equations, symplectic geometry, etc.) and physics (mechanics, geometrical optics, etc.). After giving a survey on webs in differential geometry and algebraic geometry, the book presents new results on partial differential equations, integrable systems, holomorphic dynamics and nonlinear optics obtained through web theory.

6.3 Quadratic maps6.3.1 Diffeomorphic maps; 6.3.2 Map f = x2 + c; 6.3.3 Map fyB0 = yx2 + Bx = yx2 + (b + l)x; 6.3.4 Generic quadratic map and f = x2 + px + q; 6.4 Cubic maps; 6.3.5 Families as sections; 6.4.2 Map fc = x3 + c; 6.4.4 fy = x3 + yx2; 6.4.5 Map fa; c = ax3 + (1 - a)x2 + c; 6.5 Quartic maps; 6.5.1 Map fc = x4 + c; 6.6 Maps fd; c{x) =xd + c; 6.7 Generic maps of degree d f(x) = Edi=0 aixi; 7. Conclusion; Bibliography