The Real Fatou Conjecture. (AM-144)

Jacek Graczyk
Grzegorz Świątek
Pages: 148
https://www.jstor.org/stable/j.ctt7zv8rh

1. Front Matter
(pp. i-vi)
(pp. vii-2)
3. Chapter 1 Review of Concepts
(pp. 3-24)

Quadratic polynomials from the perspective of dynamical systems. Among non-linear smooth dynamical systems quadratic polynomials are analytically the simplest. Yet, far from being trivial, they have been subject of intense research for a couple of decades. A number of difficult papers have been produced and many key questions remain unsolved. Admittedly, some phenomena that are a staple of dynamical systems, such as homoclinic intersections, are impossible in one dimension. The flip side is that the simplicity of the system makes it possible to approach rigorously phenomena that are out of reach in higher dimensions, to just name the transition to...

4. Chapter 2 Quasiconformal Gluing
(pp. 25-44)

The main objective of this chapter is to formulate Theorem 2.1 and conclude the Reduced Theorem. We will introduce a concept ofsaturated mapswhich facilitates gluing of quasiconformal branchwise equivalences. Given a pair of terminal box mappings we proceed by removing their central branches and replacing monotone branches by their filled-in versions which map onto the restrictive interval. The resulting saturated maps are quasiconformally branchwise equivalent provided the terminal box mappings were so. In the infinitely renormalizable case, we obtain infinitely many branchwise equivalent pairs of saturated maps. Our aim is to combine the branchwise equivalences into one quasiconformal...

5. Chapter 3 Polynomial-Like Property
(pp. 45-66)

Let us recall Theorem 1.1. This chapter is devoted to its proof.

Theorem 1.1Let f be a renormalizable quadratic polynomial without attracting or indifferent periodic orbits and let${{I}_{1}}\supset \cdots$be the sequence, finite or not, of its locally maximal restrictive intervals (see Definition 1.3.4). Let fidenote the first return map into IiThen, for every i, function fiis conjugate to a unimodal quadratic polynomial by an L-quasi-symmetric homeomorphism sending Iito(–1, 1).The constant L is independent of i.

Remark: In reality,Lis independent offas well, but we don’t need this fact.

Theorem...

6. Chapter 4 Linear Growth of Moduli
(pp. 67-108)

Box mappings were introduced in [13] as a tool for studying the dynamics of real unimodal polynomials. In the same paper, the main property of growing moduli was proved. This generalized earlier results obtained for certain ratios on the real line. In [14], a more general result was presented with a slightly different proof, not more complicated that the original proof of a weaker result in [13]. We state the main theorem of [14] as Theorem 1.2. The generalization consists in allowing a large class of holomorphic box mappings without any connection with real dynamics. Theorem 1.2 found already applications...

7. Chapter 5 Quasiconformal Techniques
(pp. 109-142)

The main objective of this section is to present a proof of Theorem 1.3. So we assume that unimodal polynomialsfand$\hat{f}$are real, topologically conjugate, the critical orbits omit the fixed points, and haveoddperiodic orbits on the real line. These are topological assumptions and iffsatisfies them than its both fixed points are repelling,fhas orbits with infinitely many different periods (Sharkovski’s theorem) and the first return time to the restrictive interval is greater than 2.

In the proof Theorem 1.3 a major issue is the choice of domains of analytic continuations of branches...

8. Bibliography
(pp. 143-146)
9. Index
(pp. 147-148)
10. Back Matter
(pp. 149-149)