Physics On the Invariant Sets of a Family of Quadratic ... - Project Euclid

Communications in Mathematical

Commun. Math. Phys. 88, 479-501 (1983)

Physics © Springer-Verlag 1983

On the Invariant Sets of a Family of Quadratic Maps M. F. Barnsley*, J. S. Geronimo**, and A. N. Harrington School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Abstract. The Julia set Bλ for the mapping z->(z — λ)2 is considered, where λ is a complex parameter. For λ ^ 2 a new upper bound for the Hausdorff dimension is given, and the monic polynomials orthogonal with respect to the equilibrium measure on Bλ are introduced. A method for calculating all of the polynomials is provided, and certain identities which obtain among coefficients of the threeterm recurrence relations are given. A unifying theme is the relationship between Bλ and /ί-chains λ± ]/(λ± ]/{λ± ...), which is explored for —\^λ^2 and for λe(£ with \λ\^4, with the aid of the Bδttcher equation. Then B; is shown to be a Holder continuous curve for |A|2, so that ρ oo

ρ. This implies that (ln|)/lnρ is an upper bound to the Hausdorff dimension of Bλ. Q.E.D. Brolin has given the following upper bounds for the Hausdorff dimension of

(ϋ)

?

7—77^,—-UHWTΛ

.

v a l i d

f o r

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M. F. Barnsley, J. S. Geronimo, and A. N. Harrington

Our bound improves over both of these, where they apply. We note that for λ = 5, Theorem 2 yields the upper bound 0.564 whilst (i) gives 0.636. Thus our bound is good enough, at λ = 59 to distinguish Bλ from the classical ternary set of Cantor, whose Hausdorff dimension is ln2/ln3 = 0.631, see [18]. We next give a construction, involving a particular sequence {σn(x)} of approximating distributions, for an invariant distribution σ(x) of Tλ, supported upon Bλ. Here σ(x) is an example of the equilibrium distributions described by Brolin [3, Chap. Ill], and the σw(x)'s are related to the orthogonal polynomials given in the next section. Let Kn = T^nλ, which consists of the 2" real points λ ± ]/(X ± j/(X ± ... ± ]/(I)...)) where there are n plus - or - minus signs. Let

*»(*)= i Σ θ(χ-y), L

yeKn

where θ(x) = 0 when χ5^0 and θ(x) = l when χ > 0 . Thus, σn(x) equals the proportion of members of Kn which are less than x. It is straightforward to prove that {σn(x)} converges uniformly to a continuous distribution σ(x), for x e R It is also straightforward to show, and in any case it follows from Brolin, that σ(x) provides an invariant measure under Tλ, according

\f(x)dσ{x)= E

j f(Tx)dσ(x) Ύ-^E

for all Borel measurable subsets E of R a n d all measurable functions/ When λ = 2, T2z = {z-2)2, and we have [23]

Let F denote the set of all Borel measurable subsets of Bλ. Then (Bλ, F, σ, Tλ) is a system as defined by Billingsley [6]. It is readily proved to be isomorphic to the system formed by the left-shift on Ω with the usual uniform measure. Consequently (Bλ, F, σ, Tλ) is mixing with entropy In 2. The system is also isomorphic to the one formed by z-^z2 on the unit circle in (C, with circular Lebesgue measure. This is one way to see the connection between the system which exists when λ Ξg: 2 and that which exists when λ = 0.

2.2. Orthogonal Polynomials One way of characterizing the invariant measure σ when 2 ^ /I < oo is by means of the associated set of monic orthogonal polynomials. We denote this set by {Pn(x)}™= _1? where P_ 1 ( X ) Ξ 0 . For n^O, Pn(x) has degree n and the coefficient of xn is unity. The polynomials obey \Pn{x)Pm{x)dσ{x) = 0

for

n + m.

Invariant Sets for Quadratic Maps

485

These polynomials provide an interesting generalization of the Tchebycheff 1 polynomials {Tn(x) = cos(ncos~ x)} to which, in view of the explicit formula for the measure given at the end of the last section, they must be related by when

Pn(x) = 2Tn(\x-ί)

λ = 2.

One would like to know how the invariance of the measure under Tλ relates to the structure of the polynomials. Also, what can be said about the associated threeterm recurrence relations? Let us introduce a second set of monic orthogonal polynomials {Qn(x)}™=-V where Q_1(χ) = 0. For nΞ>0, Qn(x) has degree n and the coefficient of x11 is unity. They obey jxQn(x)Qm(x)dσ(x) = O for n φ m . Then we have, for n^O and 2^A2 and α^ = 2, and (6) becomes exactly the three-term recursion relation for the Tchebycheff polynomials {TJ^x— 1)}^L _r From this it follows that the zeros of T2n{\x—ΐ) are precisely the set of numbers

1

n times

'

The densification of the latter set of numbers on [0,4] can thus be seen as an example of Blumenthal's theorem [7] on the distribution of zeros of orthogonal polynomials upon the support of the measure. Further information, which relates in particular to the sequence of approximating measures {σn(x)} given in Sect. 2.1, is obtained by examining the

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M. F. Barnsley, J. S. Geronimo, and A. N. Harrington

polynomials of the second kind, {P*(x)}^=0, which are defined by ι

+ίW

+ίW

for

P n(x)=^" _" dσ(y) / x y Theorem 5. For all ne {0,1,2,...}, 1

1

ne{- 1,0,1,2...}.

2

(9)

(10)

P 2n+1(x) = (x-λ)P n((x-λ) ) and 1 2n

' ^'

P

2n+2Wpi Γ

2n-l\X)

(x-λ)

(11)

Proof. From (9) one has

where the in variance of the measure under Tλ has been exploited. We now split up the denominator and use (4), which yields

In the second integral here we make the change of variable y — λ-> — (y — λ), use the symmetry of the measure and / about λ, and again exploit (4), to provide

From this (10) is immediate. Equation (9) also implies the recursion relation P1n+M)

= (x-λ)P1n{x)-a2n+1Pl_1{x)

for

ne{0,l,2,...},

1

with P _ί(x) = 0, and PQ(X) = 1. This implies (11) when a2+1 is eliminated with the aid of (6) wherein λ = x. Q.E.D. From (10) it is apparent, in contrast to the previous case, that the odd polynomials of the second kind are easily calculated from the even ones. Some examples of the polynomials are

2

P\(x) = {x-λ) ((x - λ) -λ) = P^P^x) = ™ P4(x),

k=o

=o

Invariant Sets for Quadratic Maps

489

Now consider the moment functions

and

G(x ) = J iχ-y From the theory of Pade approximants [2], one has that the approximant to G(x) is P

for

ln-lM(x)= ffψ,

[n—l/ή](x)

e{l,2,...}.

W

Using (4) and the fact that S2m{x) = Sm(x2 — λ) we discover the remarkable result [2n-

l/2n'](x)

l/n~]((x-λ)2)

= (x-λ)[n-

for n= 1 , 2 , 3 , . . . .

7c

Also, forrc= 2 where /ce{0,1,2,...}, we find

which makes contact with the sequence of approximating measures {σn(x)}. Finally, we note that since (Bλ, F, σ, Tλ) is a mixing system, so is (Bλ, F, σ, 7J) for ne{l,2,3,...}. Hence P2n(x) + λ provides a mixing transformation on Bλ, with respect to σ. Shifting Bλ to the left by subtracting λ, and correspondingly adjusting the measure, this shows that each of the polynomials P2n(x + λ) provides a mixing transformation upon the shifted system. This extends a result of Adler and Rivlin [1], and is itself a special case of a wide reaching theorem [5].

3. The Cases - 1 / 4 ^ A ^ 2 and U l ^ 1/4 with In this section, the Julia set Bλ is connected, and a central role is played by the Bδttcher equation, see [12], for Tλ at oo T0oHλ = HλoTλ,

Hλ(z) = z + O(l)at

oo.

(la) 1

We actually use the equivalent equation in terms of inverses, where Fλ = H^ , ?)9

Fλ(z) = z +0(1) at oo.

(lb)

We let C be the complex plane, C = Cu{oo} and D 0 = {ze(C||z|>l}. Then we shall see that Fλ maps Do conformally onto a region bounded by Bλ, and by means of Fλ we can relate λ-chains to Bλ. 3.1. Two Constructions for Bλ We present two constructions for Bλ one from the "outside", and one from the "inside", when —\^λ^2. The first is not new in principle: it involves the

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M. F. Barnsley, J. S. Geronimo, and A. N. Harrington

formation of an increasing sequence of domains, successive inverse images under Tλ of a neighborhood of oo, as suggested by Fatou [12] and by Julia [19]. We include it both for completeness and for comparison with the second method. The second construction, from the "inside", provides a decreasing sequence of domains and a corresponding sequence of functions, from Do to the domains, converging uniformly to Fλ(z) on compact subsets of Do. Of interest are the complements of the domains, which form an increasing sequence of trees, with fractal-like structure [20] and two-dimensional measure zero, which serve to describe Bλ from the interior. This construction turns out to be important: we have recently proved [4] that this sequence of trees converges to Bλ itself, for infinitely many values of Ae(0,2). For λ jg — I we define and

b= ]/\λ\ + 1/4 + μ | + 1/2.

Then a is the unique positive real numbers which obeys a = λ+ ]/α. Notice that

a^b, and λ- | / α ^ 0 , for

-^λg>2.

The following Lemma will allow us to make a concrete iterative solution of the Bottcher equation (lb). This construction is important later on in our discussion of Holder continuity. Lemma 3. For — \ ^ λ < 2, we may start from fo(z) = λ+ ybz and iteratively define analytic functions fn:D0-+(t for we {1,2,3,...} by

and they will satisfy

Sketch of Proof The lemma can be easily proved by induction. The exclusion of the interval [/I— j/α,α], which contains zero, follows from the facts that a^b and Tλ maps the excluded interval into itself. Thus, fn is well defined because 0φfn_ί(D0). That /LCDO^/OΦO) *s a simple calculation using the definition of b, and upon iteration we obtain the monotonicity of the images. Theorem 6. Let —\^λ^2. The sequence {fn} of Lemma 3 converges uniformly on compact subsets of Do to a function Fλ which obeys the inverse Bottcher equation (lb). Remark. We denote the boundary of Fλ(D0) by Bλ. The theorem says Tλ^1Bλ = Bλ. Then Bλ turns out to be the Julia set for Tλ, see [12] for example. Proof The theorem follows at once from Lemma 3 and Caratheodory's theorem on domain convergence, see Goluzin [15, p. 53]. Q.E.D. We can now set up a correspondence between A-chains and points on Bλ. Corresponding to ω = (eve2,e3,...)eΩ we define S0(ω,z) = z and, for

Invariant Sets for Quadratic Maps

491 ιθ

The value of the square root ]/w for we(C is fixed by writing w = Γe O^θ 00

fn(λ,z),

analytic in λ for λeL for each fixed zeD0. Moreover H(λ, z) = F(λ, z) for zeD0. Next consider Lim F(λ,z). For 0