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The Real Fatou Conjecture
By Jacek Graczyk, Grzegorz Swiatek
PRINCETON UNIVERSITY PRESS
Copyright © 1998 Princeton University PressAll rights reserved.
ISBN: 9780691002583
CHAPTER 1
Review of Concepts
1.1 Theory of Quadratic Polynomials
Quadratic polynomials from the perspective of dynamical systems. Among nonlinear 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 JuliaFatou 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 mideighties on an idea of treating jointly the real and complex onedimensional 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 nonspecialist, 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 polynomiallike 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 settheoretical difference between the range and the domain. Renormalization of real unimodal maps is a phenomenon when an interval (a socalled 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 polynomiallike, 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 socalled 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 realanalytic 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 instability in the real quadratic family.
Main Theorem.Let f and [??] be two real quadratic polynomials with a bounded forward critical orbits and no attracting or indifferent cycles. Then, if they are topologically conjugate, the conjugacy extends to a quasiconformal conjugacy between their analytic continuations to the complex plane.
Derivation of the Dense Hyperbolicity Theorem. We show that the Main Theorem implies the Dense Hyperbolicity Theorem. The reduction is based on three facts, two from complex dynamics and one from real, which we state here with proofs.
Fact 1.2.1Consider the set of quadratic polynomials fa (z) = az(1  z) where a is analytic parameter. For some a, let Ca [subset] C denote the set of all b such that fa and fb are conjugated on the complex plane by a quasiconformal homeomorphism. Then Ca is either {a}, or is open.
Proof:
The proof of this fact follows by the method of quasiconformal deformations introduced in [38]. The result is implicit in [27]. Suppose that fa is q.c. conjugate with fa, and a ≠ b. The conjugacy H can not be conformal on the whole plane and thus there is an fbinvariant Beltrami coefficient μ which is not identically equal to 0. We will show that b belongs to Ca together with an open ball. To this end observe that for every c [member of] C such that c < 1/[parallel]μ[parallel]∞, c x μ is an fbinvariant Beltrami coefficient. Let Hc be a solution of the Beltrami equation
dH/d[bar.z] = cμ dH/dz
normalized by the condition that 0, 1, ∞ are the fixed points. By the measurable Riemann mapping theorem, see [3], Hc depends analytically on c and fr(c) = Hc ο fb ο H1c is an analytic family of analytic functions. By topology, fr(c) is a 21 branched covering of the Riemann sphere which fixes 0 and ∞. Since ∞ is also a branching point, fr(c) is a family of quadratic polynomials. The eigenvalue e(c) := d/dz fr(c)(0) is an analytic function of c and the image of the ball c < 1/[parallel]μ[parallel]∞ by e is either a point or an open set. The first possibility is excluded since r(1) = a and r(0) = b. The function e(c) gives an analytic reparametrization e(c)z(1  z) of fr(c) ·
Fact 1.2.2There are only countably many complex values of a for which the map a >ax(1  x) has a neutral periodic point.
Proof:
We will prove a stronger statement.
If k > 0, and λ is a complex number with absolute value less or equal to 1, then the pair of equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
has only finitely many solutions (a, z).
The proof is based on the following theorem about Riemann surfaces of algebraic functions, see [9] Theorem IV.II.4 on pages 231232,
Fact 1.2.3Consider the equation P(a, z) = 0 where P is an irreducible polynomial of two complex variables. Then the set of solutions, compactified by adding points at infinity, has the structure of a compact Riemann surface. Moreover, projections on a and z are meromorphic of this surface.
This theorem applied to the polynomial fka (z)  z = 0 implies that the set of solutions splits into the union of finitely many compact Riemann surfaces. On each of these, the function
dfka/dz (z)
is meromorphic. If it takes value λ infinitely many times, by the identity principle it must be constant on one of the surfaces, call it S. If a pair (a, z) solves both equations, it means that a must be in the connectedness locus in the parameter space, and z is in the filled Julia set. Hence, both projections map the finite points of S into a bounded set in the complex plane. The image of S under either projection must be compact, since the projection is continuous. But since the projections are also open mappings or constant, the image of either of them is just a point. Hence, S must be a point, which is impossible.
Fact 1.2.4Two real quadratic polynomials f and g with bounded critical orbits and such that f has no attracting periodic orbit, normalized so that their critical points zf and zg, respectively, are maxima, are topologically conjugate on the real line if and only if for every n > 0 both differences fn (zf)  zf and gn (zg)  zg have the same sign.
Proof:
The "only if" part is obvious and the rest is contained in Theorem 2.10 in [17].
Suppose that fa(x) = ax(1  x) has no stable periodic orbits. Let Ta denote the set of parameter values b such that fb(x) = bx(1  x) is topologically conjugate to fa on the real line. Then Ta is closed. Indeed, if a sequence bn [member of] Ta converges to b, the signs of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] remain fixed for all k. By continuity, the differences fkb(1/2)–1/2 either remain of the same sign, in which case our assertion follows from Fact 1.2.4, or some may vanish. If one of them vanishes, it means that 1/2 is periodic by fb. But then the implicit function theorem implies that for all parameters in a neighborhood of b there is a stable periodic orbit. This is a contradiction since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all n have no such orbits.
Now Fact 1.2.1 means that for any a [member of] (0, 4] such that fa has only repelling periodic orbits, the quasiconformal class Ca intersected with the real line is either a point or is open. The Main Theorem means that Ca [intersection] R = Ta. Since Ta is closed, it must be a point.
To prove the Dense Hyperbolicity Theorem, let a [member of] (0, 4]. In view of Fact 1.2.2, we can assume without loss of generality that fa has only repelling periodic orbits. In every neighborhood of a we can find a1 ≠ a so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] still only has repelling periodic orbits. As observed in the preceding paragraph, fa and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are not topologically conjugate. In view of Fact 1.2.4 this means that for some k the differences fka(1/2)  1/2 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1/2)  1/2 have different signs. By the intermediate value theorem, for some a0 between a1 and a we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This means that 1/2 belongs to an attracting periodic orbit and proves the Dense Hyperbolicity Theorem.
Historical notes. The Dense Hyperbolicity Conjecture has had a long history. In a paper from 1920, see [10], Fatou expressed the belief that "general" (generic in today's language?) rational maps are expanding on the Julia set. Our result may be regarded as progress in the verification of his conjecture. More recently, the fundamental work of Milnor and Thurston, see [32], showed the monotonicity of the kneading invariant in the quadratic family. They also conjectured that the set of parameter values for which attractive periodic orbits exist is dense, which means that the kneading sequence is strictly increasing unless it is periodic. The Dense Hyperbolicity Theorem implies Milnor and Thurston's conjecture. Otherwise, we would have an interval in the parameter space filled with polynomials with an aperiodic kneading sequence, in violation of the Dense Hyperbolicity Theorem.
Yoccoz, [41], proved that a nonhyperbolic quadratic polynomial with a fixed nonperiodic kneading sequence is unique up to an affine conjugacy unless it is infinitely renormalizable. This implied our Main Theorem in all cases except for the infinitely renormalizable. His method is different from one explained in this book and the Main Theorem does not even appear as a step in the proof. Instead, geometric estimates are established in the phase space and then used in the parameter space to explicitly show that the set Ta is a point.
In the infinitely renormalizable case, the work of [39] proved the Main Theorem for infinitely renormalizable polynomials of bounded combinatorial type. The paper [24] achieved the same for some infinitely renormalizable quadratic polynomials not covered by [39]. This book follows the method of [16].
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