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The Real Fatou Conjecture. (AM-144)

The Real Fatou Conjecture. (AM-144)

Jacek Graczyk
Grzegorz Świątek
Copyright Date: 1998
Pages: 148
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  • Book Info
    The Real Fatou Conjecture. (AM-144)
    Book Description:

    In 1920, Pierre Fatou expressed the conjecture that--except for special cases--all critical points of a rational map of the Riemann sphere tend to periodic orbits under iteration. This conjecture remains the main open problem in the dynamics of iterated maps. For the logistic family x- ax(1-x), it can be interpreted to mean that for a dense set of parameters "a," an attracting periodic orbit exists. The same question appears naturally in science, where the logistic family is used to construct models in physics, ecology, and economics.

    In this book, Jacek Graczyk and Grzegorz Swiatek provide a rigorous proof of the Real Fatou Conjecture. In spite of the apparently elementary nature of the problem, its solution requires advanced tools of complex analysis. The authors have written a self-contained and complete version of the argument, accessible to someone with no knowledge of complex dynamics and only basic familiarity with interval maps. The book will thus be useful to specialists in real dynamics as well as to graduate students.

    eISBN: 978-1-4008-6518-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (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.


  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)