Complex Dynamics and Renormalization

Title page
1 Introduction
1.1 Complex dynamics
1.2 Central conjectures
1.3 Summary of contents
2 Background in conformal geometry
2.1 The modulus of an annulus
2.2 The hyperbolic metric
2.3 Metric aspects of annuli
2.4 Univalent maps
2.5 Normal families
2.6 Quasiconformal maps
2.7 Measurable sets
2.8 Absolute area zero
2.9 The collar theorem
2.10 The complex shortest interval argument
2.11 Controlling holomorphic contraction
3 Dynamics of rational maps
3.1 The Julia and Fatou sets
3.2 Expansion
3.3 Ergodicity
3.4 Hyperbolicity
3.5 Invariant line £ields and complex tori
4 Holomorphic motions and the Mandelbrot set
4.1 Stability of rational maps
4.2 The Mandelbrot set
5 Compactness in holomorphic dynamics
5.1 Convergence of Riemann mappings
5.2 Proper maps
5.3 Polynomial-like maps
5.4 Intersecting polynomial-like maps
5.5 Polynomial-like maps inside proper maps
5.6 Univalent line £ields
6 Polynomials and external rays
6.1 Accessibility
6.2 Polynomials
6.3 Eventual surjectivity
6.4 Laminations
7 Renormalization
7.1 Quadratic polynomials
7.2 Small Julia sets meeting at periodic points
7.3 Simple renormalization
7.4 Examples
8 Puzzles and infinite renormalization
8.1 In£inite renormalization
8.2 The Yoccoz jigsaw puzzle
8.3 In£inite simple renormalization
8.4 Measure and local connectivity
8.5 Laminations and tableaux
9 Robustness
9.1 Simple loops around the postcritical set
9.2 Area of the postcritical set
10 Limits of renormalization
10.1 Unbranched renormalization
10.2 Polynomial-like limits of renormalization
10.3 Proper limits of renormalization
10.4 Extracting a univalent line £ield
11 Real quadratic polynomials
11.1 Intervals and gaps
11.2 Real robustness
11.3 Corollaries and generalizations
A Orbifolds
A.1 Smooth and complex orbifolds
A.2 Coverings and uniformization
A.3 The orbifold of a rational map
B A closing lemma for rational maps
B.1 Quotients of branched coverings
B.2 Critically £inite rational maps
B.3 Siegel disks, Herman rings and curve systems
B.4 Rational quotients
B.5 Quotients and renormalization
Bibliography
Index