Contents -- Preface -- 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. Fragments of theory -- 4.1 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 -- 4.8 Combinatorics of discriminants and resultants -- 4.9 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).

5.1.2 Orbits of order two -- 5.1.3 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 -- 6.3 Quadratic maps -- 6.3.1 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 -- 6.3.5 Families as sections -- 6.4 Cubic maps -- 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.