Well-Approximable Points for Julia Sets with Parabolic ... - CiteSeerX

Computational Methods and Function Theory Volume 1 (2001), No. 1, 89–97

Well-Approximable Points for Julia Sets with Parabolic and Critical Points Bernd O. Stratmann, Mariusz Urba´ nski, and Michel Zinsmeister Abstract. In this paper we consider rational functions f : C → C with parabolic and critical points contained in their Julia sets J(f ) such that ∞ X |(f n )0 (f (c))|−1 < ∞ n=1

for each critical point c ∈ J(f ). We calculate the Hausdorff dimensions of subsets of J(f ) consisting of elements z for which inf{dist(f n (z), Crit(f )) : n ≥ 0} > 0 and which are well-approximable by backward iterates of the parabolic periodic points of f . Keywords. Rational functions, critical phenomena, ergodic theory, fractal geometry, Diophantine analysis. 2000 MSC. 37A45, 30C99, 30D40, 11K36.

1. Introduction and statement of main results In [8], [9], [10], [11], and [12] we developed a certain type of Diophantine analysis for geometrically finite Kleinian groups with parabolic elements, parabolic rational maps and tame parabolic iterated function systems. In this paper we extend this analysis to rational maps f : C → C with parabolic points and with Julia sets J(f ) containing critical points such that for each of these critical points c ∈ J(f ) we have that ∞ X |(f n )0 (f (c))|−1 < ∞. n=1

The idea is to perform this Diophantine analysis on the set of points z ∈ J(f ) for which we have, with Crit(f ) denoting the set of critical points of f , that inf{dist(f n (z), Crit(f )) : n ≥ 0} > 0.

Received September 17, 2001. M.Z. was supported partially by the NSF Grant DMS 9801583. c 2001 Heldermann Verlag ISSN 1617-9447/$ 2.50 °

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We give a further development of the techniques elaborated in the afore-mentioned papers, mainly by using results obtained in [2] and [4] (see also [5]). In particular, a crucial quantity in our main theorem (Theorem 1.1) is the so-called dynamical dimension DD(J(f )) of J(f ), in the literature occasionally also referred to as the essential dimension, which is given by ([2]; see also [4], [5]) DD(J(f )) := sup{HD(µ)}. Here, the supremum is taken with respect to all f -invariant, ergodic Borel probability measures µ of positive entropy, and HD(µ) denotes the Hausdorff dimension of µ. In [2] it was shown that DD(J(f )) coincides with the least possible exponent for conformal measures associated with f . Also, in [5] we obtained that DD(J(f )) coincides with the hyperbolic dimension of f in the sense of Shishikura [7]. In order to state our main result, we first have to introduce some notation. Let Ω be the set of all parabolic periodic points of f , that is Ω := {ω ∈ C : f q (ω) = ω and (f q )0 (ω) = 1 for some q ≥ 1}. It is well-known that Ω S is a finite subset of J(f ). For fixed ω ∈ Ω and for every pre-parabolic point x ∈ n≥0 f −n (ω), let n(x) denote the least integer such that f n(x) (x) = ω. Since the set of critical points of f is finite and since critical points are not periodic, there exists a number q(ω) ≥ 2 such that for every pre-parabolic point x for which n(x) ≥ q(ω), we have that [ Crit(f ) ∩ f −n (x) = ∅. n≥0

In the following let x = x(ω) denote some fixed element of this type. Then for S every y ∈ n≥0 f −n (x) there exists a unique integer k(y) such that f k(y) (y) = x. If B(w, r) denotes the closed ball centred at w of radius r, then for some fixed ρ0 > 0 and every κ > 0 we define ³ ¡ ¢1+κ ´ Byκ := B y, ρ0 |(f k(y) )0 (y)|−1 . Furthermore, for ² > 0 let

κ Jx,² :=

\[[

Bzκ ,

q≥1 n≥q ∗

where the union

S

∗

is taken with respect to all elements [ [ z∈ f −l (x) \ f −j (B(Crit(f ), ²)) l≥0

j≥0

for which k(z) = n. Our main interest in this paper will be focused on the sets [ [ [ κ Jxκ := Jx,² , Jωκ := Jzκ and J κ := Jωκ . ²>0

{z: n(z)=q(ω)}

ω∈Ω

We are now in the position to state the main result of this paper. Here, we have used the common notation p(ω) to denote the number of petals of ω ∈ Ω.

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Theorem 1.1. Let f : C → CPbe a rational map with parabolic periodic points ∞ n 0 −1 < ∞ for each critical point and critical points such that n=1 |(f ) (f (c))| c ∈ J(f ). Then for every parabolic point ω ∈ Ω and each κ > 0 it holds that  DD(J(f ))   if κ ≥ DD(J(f )) − 1   1+κ HD(Jωκ ) =    DD(J(f )) + κp(ω) if κ ≤ DD(J(f )) − 1.  1 + κ(1 + p(ω)) Hence in particular, we have with pmin := min{p(ω) : ω ∈ Ω} that  DD(J(f ))   if κ ≥ DD(J(f )) − 1   1+κ HD(J κ ) =  DD(J(f )) + κpmin   if κ ≤ DD(J(f )) − 1.  1 + κ(1 + pmin )

We remark that a similar type of Diophantine analysis could also be given for repelling periodic orbits rather than parabolic periodic points. In fact, in that case the arguments and also the results would be far less involved, mainly due to the lack of ‘conformal fluctuation’ of the conformal measure at repelling periodic orbits. Finally, we remark that Theorem 1.1 represents a generalization of the following classical result in metrical Diophantine analysis by Jarn´ık [3] and Besicovitch [1] on well-approximable irrationals. ¯ ¯ ¾¶ µ½ ¯ 1 1 p ¯¯ ¯ = . HD ξ ∈ R : ¯ξ − ¯ < 2(1+σ) for infinitely many (p, q) = 1 q q 1+σ

2. Proof of Theorem 1.1

In this section we give the proof of Theorem 1.1. It will be prepared first by giving a series of separate statements. From now on we assume that the preliminaries in Theorem 1.1 are satisfied. For ² > 0, we define the set [ K² := J(f ) \ f −n (B(Crit(f ), ²)). n≥0

Observe that by [6] (Appendix B, Theorem B.1), for each ², θ > 0 there exists τ² (θ) > 0 with the property that if z ∈ K² such that f n (z) ∈ / B(Ω, θ), then for every 0 ≤ k ≤ n we have that ¡ ¢ diam f k (Cn (z, f n (z), 2τ² (θ)) < ².

Here Cn (z, f n (z), 2τ² (θ)) is the connected component of f −n (B(f n (z), 2τ² (θ))) which contains z. Note that this implies in particular that f k (Cn (z, f n (z), 2τ² (θ))) ∩ Crit(f ) = ∅. This observation is summarized in the following lemma.

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Lemma 2.1. For every ² > 0 and every θ > 0 there exists τ² (θ) > 0 such that if z ∈ K² and f n (z) ∈ / B(Ω, θ), then Cn (z, f n (z), 2τ² (θ)) ∩ Crit(f n ) = ∅. The next step in the proof of Theorem 1.1 is to verify the following upper bound for the Hausdorff dimension. S Lemma 2.2. For each ω ∈ Ω, x ∈ n≥0 f −n (ω) such that n(x) ≥ q(ω), and for every ², κ > 0 we have that ¾ ½ DD(J(f )) DD(J(f )) + κp(ω) κ . , HD(Jx,² ) ≤ min 1+κ 1 + κ(1 + p(ω)) Proof. Throughout we shall always assume that θ = dist(x, Ω). In view of Lemma 2.1, for every n ≥ 0 and each z ∈ f −n (x) ∩ K² there exists a unique holomorphic branch fz−n : B(x, 2τ² (θ)) → C of f −n which maps x to z. Since the family {fz−n : B(x, 2τ² (θ)) → C : n ≥ 0, z ∈ f −n (x) ∩ K² } is normal and since x ∈ J(f ), all the limit functions of this family are constant. Thus, it follows that (1)

lim sup{|(fz−n )0 (y)| : y ∈ B(x, τ² (θ))} = 0.

n→∞

Hence in particular, we have that M := sup{|(fz−n )0 (y)| : n ≥ 0, y ∈ B(x, τ² (θ))} < ∞, and also, that there exists s ≥ 1 sufficiently large such that |(f n )0 (z)| > 2K 2 for all n > s and z ∈ f −n (x) ∩ K² . Here K ≥ 1 refers to the Koebe constant corresponding to the scale 1/2. For every l ≥ 0 we define ( [ Zl := z ∈ f −n (x) ∩ K² : n≥0

M 2(l+1)

1 M ≤ ≤ l k(z) 0 |(f ) (z)| 2

)

.

We claim that for each l ≥ 0 the family DD(J(f )). Using the latter estimate, we first make the following computation. X X XX |(f k(z) )0 (z)|−t Σt := |(f n )0 (z)|−t = n≥1 z∈K² ∩f −n (x)

=

XX

l≥0 z∈Zl

|(f k(z) )0 (z)|− DD(J(f )) |(f k(z) )0 (z)|−(t−DD(J(f )))

l≥0 z∈Zl

(2) ≤

XX

|(f k(z) )0 (z)|− DD(J(f )) M t−DD(J(f )) 2−l(t−DD(J(f )))

l≥0 z∈Zl

¿ sM t−DD(J(f )) K DD(J(f ))

X

2−l(t−DD(J(f ))) < ∞.

l≥0

Now observe that for each k ≥ 1, the family {Bzκ : z ∈ K² ∩ f −n (x), n ≥ k} κ represents a covering of Jx,² , and that for k increasing the upper bound for the radii of this covering tends to zero. This follows since by (1) we have

lim max{|(f n )0 (z)| : z ∈ K² ∩ f −n (x)} = ∞.

n→∞

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Furthermore, by the computation in (2), for every k ≥ 1 we have for the radii of the elements of this covering that X X X X ¡ ¢t/(1+κ) (ρ0 |(f n )0 (z)|−1 )1+κ ¿ |(f n )0 (z)|−t n≥k z∈K² ∩f −n (x)

n≥k z∈K² ∩f −n (x)

= Σt < ∞.

This immediately gives for the t/(1 + κ)-dimensional Hausdorff measure H t/(1+κ) κ κ that Ht/(1+κ) (Jx,² ) < ∞, and hence that HD(Jx,² ) ≤ t/(1 + κ). By letting t tend to DD(J(f )), we obtain DD(J(f )) κ . (3) HD(Jx,² )≤ 1+κ In order to obtain the second upper estimate of the lemma, note that Fatou’s flower theorem implies that for each n ≥ 1 and z ∈ K² ∩ f −n (x) the ball B(ω, (ρ0 |(f n )0 (z)|−1 )κ ) admits a covering by balls of radii (ρ0 |(f n )0 (z)|−1 )κ(p(ω)+1) such that the cardinality of this covering is comparable to |(f n )0 (z)|κp(ω) . Consequently, by Lemma 2.1 and Koebe’s distortion theorem, each ball Bzκ with z ∈ K² ∩f −n (x) can be covered by at most a number comparable to |(f n )0 (z)|κp(ω) of balls with radii comparable to (ρ0 |(f n )0 (z)|−1 )1+κ(1+p(ω)) . Hence, using this observation and (2), we obtain for each λ > 0 that H

DD(J(f ))+κp(ω) +λ 1+κ(1+p(ω))

¿ lim

k→∞

¿ lim

k→∞

κ (Jx,² ) X X

(ρ0 |(f n )0 (z)|−1 )(1+κ(1+p(ω)))(

DD(J(f ))+κp(ω) +λ 1+κ(1+p(ω))

) |(f n )0 (z)|κp(ω)

n≥k z∈K² ∩f −n (x)

X

X

|(f n )0 (z)|−(DD(J(f ))+λ(1+κ(1+p(ω)))) = 0.

n≥k z∈K² ∩f −n (x)

This implies that

DD(J(f )) + κp(ω) + λ. 1 + κ(1 + p(ω)) Hence, by letting λ tend to 0, we conclude that DD(J(f )) + κp(ω) κ HD(Jx,² )≤ . 1 + κ(1 + p(ω)) κ HD(Jx,² )≤

As an immediate consequence of this lemma we have the following corollary. Corollary 2.3. For each ω ∈ Ω and for every κ > 0 we have that ½ ¾ DD(J(f )) DD(J(f )) + κp(ω) κ , HD(Jω ) ≤ min 1+κ 1 + κ(1 + p(ω)) and ½ ¾ DD(J(f )) DD(J(f )) + κpmin HD(Jκ ) ≤ min , . 1+κ 1 + κ(1 + pmin )

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For the remaining part of the proof of Theorem 1.1, we assume additionally that θ > 0 is chosen so small that for every z ∈ (J(f ) \ Ω) ∩ B(Ω, θ)Sthere exists n ≥ 1 with f n (z) ∈ / B(Ω, θ). For fixed ² > 0 and for z ∈ K² \ n≥0 f −n (Ω) let {nj }∞ j=1 denote the increasing sequence of natural numbers which is given by nj f (z) ∈ / B(Ω, θ) for all j, and f k (z) ∈ B(Ω, θ) for all k ∈ / {n1 , n2 , . . .}. Clearly, such a sequence is infinite. Also, for every j ≥ 1 we let rj (z) := |(f nj )0 (z)|−1 , and for every r > 0 sufficiently small we define rmin (z) := min{rk (z) : rk (z) > r} and rmax (z) := max{rk (z) : rk (z) ≤ r}. Furthermore with x = x(ω) chosen as before for ω ∈ Ω, if z ∈ K² ∩ f −n (x) for some n ≥ 0, then we define 1

rω (z) :=

|(f n )0 (z)|

.

For each ² > 0 sufficiently small such that K² 6= ∅, we obtained in [2] that there exists a Borel probability measure m² supported on K² and a number h² ≤ DD(J(f )) such that the following properties are satisfied. • (4)

lim h² = DD(J(f )); ²&0

• for every Borel set E ⊂ J(f ) such that f |E is injective, we have that Z (5) m² (f (E)) ≥ |f 0 (z)|h² dm² (z); E

• for every Borel set F ⊂ K² \ ∂B(Crit(f ), ²) such that f |F is injective, we have that Z m² (f (F )) = |f 0 (z)|h² dm² (z). F

Finally, the conformal fluctuation function ζ² of m² is defined for z ∈ K² and r > 0 by ζ² (z, r) :=

m² (B(z, r)) . r h²

Now, the remaining part of the proof of Theorem 1.1 is completely analogous to the constructions in [12], and we refer to this paper for the details. More precisely, in order to obtain uniform estimates for the conformal fluctuation function, we can now proceed as in section 3 of [12]. Note that the proof of these estimates uses (5), Lemma 2.1, Corollary 2.3 and the local properties of f around parabolic points.

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S Lemma 2.4. If z ∈ K² \ n≥0 f −n (Ω) and r > 0 such that rmin (z) = rj (z) and f nj +1 (z) ∈ B(ω, θ) for some j ∈ N and ω ∈ Ω, then we have that µ ¶(h² −1)p(ω) µ ¶1/(p(ω)+1) rmax (z) r   for rmin (z) ≥ r ≥ rmin (z) ,    rmin (z) rmin (z) ζ² (z, r) ¿ ¶h −1 ¶1/(p(ω)+1) µ µ    rmax (z) rmax (z) ²   for rmax (z) ≤ r ≤ rmin (z) . r rmin (z)

If z ∈ K² ∩ f −n (x) for some n ≥ 0 and ω ∈ Ω, then we have for all 0 < r ≤ rω (z) that ζ² (z, r) ³ r (h² −1)p(ω) . Furthermore, on the basis of Lemma 2.1, Lemma 2.4 and the local properties of f around parabolic points, we can then proceed as in section 4.1 of [12] to obtain the following result. Note that in [12], in order to derive these lower estimates for the Hausdorff dimension, we employed precisely the type of estimates for the conformal fluctuation function, which we have just derived for ζ² in Lemma 2.4 above. Lemma 2.5. For each ω ∈ Ω and every ² > 0 we have that  h²   if κ ≥ h² − 1,  1 + κ κ HD(Jω ) ≥  h² + κp(ω)   if κ ≤ h² − 1.  1 + κ(1 + p(ω)) Consequently, it follows that  h²    1 + κ HD(J κ ) ≥  h² + κpmin    1 + κ(1 + pmin )

if κ ≥ h² − 1, if κ ≤ h² − 1.

Finally, the proof of Theorem 1.1 now follows from Lemma 2.2 and Lemma 2.5, where we let ² tend to 0 and use the limit behaviour as stated in (4).

Acknowledgement. A substantial part of the research for this paper was done at the Mathematics Department of the University of Orleans (France). We should like to thank this institution for its warm hospitality and excellent working conditions. Also, we should like to thank the referee for his/her careful reading of the paper.

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References 1. A. S. Besicovitch, Sets of fractional dimension(IV): On rational approximation to real numbers, Journ. London Math. Soc. 9 (1934), 126–131. 2. M. Denker and M. Urba´ nski, On Sullivan’s conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), 365–384. 3. V. Jarn´ık, Diophantische Approximationen und Hausdorff Maß, Mathematicheskii Sbornik 36 (1929), 371–382. 4. F. Przytycki, Conical limit set and Poincar´e exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999), 2081–2099. 5. F. Przytycki and M. Urba´ nski, Fractals in the Plane-Ergodic Theory Methods, to appear in Cambr. Univ. Press; available on http://www.math.unt.edu/∼urbanski. 6. F. Przytycki and M. Urba´ nski, Porosity of Julia sets of non-recurrent and parabolic ColletEckmann rational functions, Ann. Acad. Sci. Fenn. 26 (2001), 125–154. 7. M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. Math. 147 (1998), 225–267. 8. B. O. Stratmann, Fractal dimensions for Jarn´ık limit sets; the semi-classical approach, Ark. f¨or Mat. 33 (1995), 385–403. 9. B. O. Stratmann, Weak singularity spectra of the Patterson measure for geometrically finite Kleinian groups with parabolic elements, Michigan Math. J. 46 (1999), 573–587. 10. B. O. Stratmann and M. Urba´ nski, Jarn´ık and Julia; a Diophantine analysis for parabolic rational maps, preprint in Mathematica Gottingensis 02 (1999); to appear in Math. Scan. (2003). 11. B. O. Stratmann and M. Urba´ nski, The geometry of conformal measures for parabolic rational maps, Math. Proc. Cambr. Phil. Soc. 128 (2000), 141–156. 12. B. O. Stratmann and M. Urba´ nski, Metrical Diophantine analysis for tame parabolic iterated function systems, submitted, preprint in Mathematica Gottingensis 11 (2000), 1–30. Bernd O. Stratmann E-mail: bos@@maths.st-and.ac.uk Address: Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Scotland URL: http://www.maths.st-and.ac.uk/∼bos Mariusz Urba´ nski E-mail: urbanski@@unt.edu Address: Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA URL: http://www.math.unt.edu/∼urbanski Michel Zinsmeister E-mail: Michel.Zinsmeister@@labomath.univ-orleans.fr Address: MAPMO, Math´ematiques, Universite d’Orleans, BP 6759 45067 Orl´eans Cedex, France URL: http://www.labomath.univ-orleans.fr/descriptions/zins