Review of Concepts

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 chaos. For one reason or another, a number of mathematicians became interested in the very narrow field of quadratic polynomials.

Iteration of a quadratic polynomial leads to polynomials of progressively higher degrees and here the transparent simplicity of the system is lost. Given a polynomial of degree 2100, how does one tell that it is an iteration of a quadratic; if so how can one exploit this fact dynamically? In real dynamics, a property of quadratic polynomials which is inherited under iteration is negative Schwarzian derivative. An impressive technique has been developed based on this property, see [30]. However, specific properties of quadratic polynomials and their iterations become more evident if they are viewed as mappings of the complex plane. The classical Julia-Fatou theory provides new insights. A powerful new tool known as quasiconformal deformations becomes available. If two maps (say polynomials) are quasiconformally conjugated, one can perturb the conjugacy in such a way that a holomorphic family of conjugated systems of the same type (polynomials of the same degree) interpolating between the original ones is formed, see [39]. Nothing like this exists in the real theory. A polynomial can be perturbed explicitly by changing a parameter, but trying to manipulate the conjugacy between two real polynomials will lead to more complicated transformations, usually no more than continuous. And so from the mid-eighties on an idea of treating jointly the real and complex one-dimensional systems (see [39]) became increasingly popular. Real polynomials are right on the borderline and naturally became the proving ground for this concept.

The divide between real and complex dynamics. However, the merging of real and complex dynamics also encountered serious hurdles. The methods and the style of papers in both fields are different. In interval dynamics proofs are mostly long sequences of inequalities. To check a proof, one goes through all the inequalities and an occasional combinatorial lemma. The holomorphic dynamics is made of different ingredients. In many papers, there are few inequalities or formulas to go by. The proofs are made of concepts, often quite geometric in nature. For a non-specialist, checking a proof may present a formidable difficulty, since the key concepts are not easily put down as definitions or theorems.

A proof of Fatou's conjecture for real quadratic polynomials relies both on real and complex methods. However, the gist of many technical arguments is shifted from the real line to the complex plane. The relation between real and complex methods deserves to be carefully explained. This does not mean that we attempt to develop philosophical principles or heuristic arguments which even if widely accepted remain beyond the domain of mathematical proof. We simply try to formulate this relation rigorously.

The content of this book. In this book, our ambition is to present the proofs in a rigorous way accessible to the wide audience in dynamical systems and beyond. Hence, it is not to present all that is known about quadratic polynomials. For that, the most comprehensive source remains. We skipped the complex case and concentrate on the proof of Fatou's conjecture for real quadratic polynomials. Such a limited approach gives our work a good logical structure, allows the presentation of a wide array of concepts, and best serves our goal of making a rigorous presentation.

1.1.1 Weak hyperbolicity of quadratic polynomials

There are two properties of real quadratic polynomials that make this proof work. The first is known as "complex bounds" of renormalization. A quadratic polynomial is generally not expanding, but it always stretches sets in the large scale. The meaning of this "large scale" expansion is explained in Douady and Hubbard's definition of a polynomial-like map (see [8]): namely that the domain of the mapping, assumed to be a topological disk is mapped on a strictly larger region, which contains the closure of the original domain. The "strength" of this expansion can be measured by the width of the set-theoretical difference between the range and the domain. Renormalization of real unimodal maps is a phenomenon when an interval (a so-called restrictive interval) is mapped into itself by an iterate and this transformation is unimodal. If the original system was a quadratic polynomial, this first return map is a polynomial of high degree. The complex bounds property says that if a topological disk is suitably chosen around the restrictive interval, then the first return map becomes polynomial-like, with only one critical point in its domain, and with "strength" bounded away from 0.

The second property is related to the concept of inducing. In 1981 Michael Jakobson proved the existence of invariant measures for a large set of unimodal maps. His method was based on replacing the original mapping on pieces of the domain by iterations. In the end, he obtained a map defined almost everywhere, with infinitely many branches each being an iterate of the original transformation, all monotone, expanding and mapping onto a fixed interval. Later research showed that this property was quite prevalent. However, it cannot hold for infinitely renormalizable mappings for topological reasons. Nevertheless, even for those it remains true that high iterations become expanding, if chosen appropriately. To fully exploit this phenomenon, in 1993 we introduced a class of so-called box mappings, see [13]. In the language of box mappings, the property becomes the increase of certain conformal moduli.

It should be emphasized that the only case when both properties are satisfied is the real quadratic family. The first property belongs to real systems and is true for unimodal polynomials of any degree with generalizations to real-analytic mappings, see [26]. However, it has no counterpart for complex quadratic polynomials, see [34]. The second property is not applicable in general if the degree is greater than 2.

1.2. Dense Hyperbolicity

1.2.1 Theorem and its consequences

The Dense Hyperbolicity Theorem.In the real quadratic family

fa (x) = ax (1 - x), 0 < a ≤ 4

the mapping fa has an attracting cycle, and thus is hyperbolic, for an open and dense set of parameters a.

The Dense Hyperbolicity Theorem follows from the Main Theorem which gives an analytically checkable condition for...