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1 Open Research Online The Open University s repository of research publications and other research outputs Functions of genus zero for which the fast escaping set has Hausdorff dimension two Journal Item How to cite: Sixsmith, D J (205) Functions of genus zero for which the fast escaping set has Hausdorff dimension two Proceedings of the American Mathematical Society, 43(6) pp For guidance on citations see FAQs c 205 American Mathematical Society Version: Accepted Manuscript Link(s) to article on publisher s website: Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyright owners For more information on Open Research Online s data policy on reuse of materials please consult the policies page oroopenacuk

2 FUNCTIONS OF GENUS ZERO FOR WHICH THE FAST ESCAPING SET HAS HAUSDORFF DIMENSION TWO D J SIXSMITH Abstract We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane In general these functions lie outside the Eremenko-Lyubich class We show that for functions in this family the fast escaping set has Hausdorff dimension equal to two Introduction Suppose that f : C C is a transcendental entire function The Fatou set F (f) is defined as the set of points z C such that (f n ) n N is ormal family in a neighbourhood of z Since F (f) is open, it consists of at most countably many connected components which are called Fatou components The Julia set J(f) is the complement in C of F (f) An introduction to the properties of these sets was given in [3] For a general transcendental entire function the escaping set I(f) = {z : f n (z) as n } was studied in [7] The set I(f) now plays a key role in the study of complex dynamics, particularly with regard to the major open question, first asked in [7], of whether I(f) necessarily has no bounded components This question is known as Eremenko s conjecture The fast escaping set, A(f) I(f), was introduced in [4], and was defined in [5] by A(f) = {z : there exists l N such that f n+l (z) M n (R, f), for n N} Here, the maximum modulus function M(r, f) = max z =r f(z), for r 0, M n (r, f) denotes repeated iteration of M(r, f) with respect to the variable r, and R > 0 is such that M(r, f) > r, for r R The set A(f) also now plays a key role in the study of complex dynamics One reason for this is that A(f) has no bounded components [4, Theorem ], and this provides a partial answer to Eremenko s conjecture We refer to [5] for a detailed account of the properties of this set The Eremenko-Lyubich class, B, is defined as the class of transcendental entire functions for which the set of singular values is bounded Most existing results on the size of the Julia set of a transcendental entire function, f, concern the case that f B A key paper which concerns the size of the Julia set of a transcendental entire function which may be outside the class B is that of Bergweiler and Karpińska [5] 200 Mathematics Subject Classification Primary 37F0; Secondary 30D05 The author was supported by Engineering and Physical Sciences Research Council grant EP/J02260/

3 2 D J SIXSMITH Their principle result is as follows For E C, we let dim H E denote the Hausdorff dimension of E, and refer to [9] for a definition of Hausdorff dimension Theorem Suppose that f is a transcendental entire function and that there exist A, B, C, r > such that () A log M(r, f) log M(Cr, f) B log M(r, f), for r r Then dim H J(f) I(f) = 2 The techniques used to prove Theorem are very different to those used in earlier papers on dimension Roughly speaking, the key idea is to construct a set in which the logarithmic derivative is large, and approximately constant in certain small discs Bergweiler and Karpińska showed that the existence of this set follows from (), which imposes a very strong regularity condition on the growth of the function The aim of this paper is to show that there is a large family of functions, which need not satisfy this regularity condition, for which a set with somewhat similar properties may be constructed We then show that for functions in this family the fast escaping set has Hausdorff dimension equal to 2 In particular, we study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a halfplane In general these functions lie outside the class B For example, functions in the class B have order at least 2 (see [] or [3, Lemma 35], for instance), whereas the functions we consider may have any order less than We refer to [2] for the definitions of genus and order Our main result is as follows Here, and throughout the paper, we denote by arg(z) the principle argument of z, for z 0; in other words arg(z) ( π, π] Theorem 2 Suppose that f is a transcendental entire function of the form (2) ( f(z) = cz q + z ), where c 0, q {0,, }, and 0 < a a 2 n= Suppose that there exist positive constants θ and θ 2, such that 0 θ 2 θ < π, and also N 0 N such that (3) arg( ) [θ, θ 2 ] mod 2π, for n N 0 Suppose also that f has no multiply connected Fatou components Then, dim H J(f) A(f) = 2 Functions for which () is satisfied have no multiply connected Fatou components [5, Theorem 45] It is well-known that there are functions of the form (2), such that (3) is satisfied and which have a multiply connected Fatou component However, if U is a multiply connected Fatou component of f, then U A(f); see [4, Theorem 2] and [5, Theorem 44] We deduce the following corollary of Theorem 2 Corollary Suppose that f is a transcendental entire function of the form (2), such that (3) is satisfied Then dim H A(f) = 2

4 FUNCTIONS FOR WHICH THE FAST ESCAPING SET HAS DIMENSION TWO 3 If a transcendental entire function, f, of the form (2), also satisfies the condition (4) lim sup log + n log =, then it follows from [6, Theorem 2] that f has no multiply connected Fatou components Hence if (3) and (4) are both satisfied, then dim H J(f) A(f) = 2 Remark An example of a transcendental entire function, f, of the form (2), such that (3) is satisfied, A(f) F (f) and f has no multiply connected Fatou components was given in [7] This function is outside the class B Remark 2 If f B, then f does not have a multiply connected Fatou component [8, Proposition 3] It follows that if f B is of the form (2), and such that (3) is satisfied, then dim H J(f) A(f) = 2 An example of such a function [, p75] is cos z = n= ( z π 2 (n 2 )2 Remark 3 Rippon and Stallard [6] also studied a family of functions of genus zero In particular, they studied the dynamics of transcendental entire functions of the form ( f(z) = cz q + z ), where c R\{0}, q {0,, }, and 0 < α α 2 α n n= It follows from Theorem 2, together with results in [6], that if f is a function of this form, which also satisfies an additional condition on the minimum modulus of f, then J(f) and J(f) I(f) are spiders webs of Hausdorff dimension 2; we refer to [6] for background and definitions The structure of this paper is as follows First, in Section 2, we give umber of definitions which are used throughout the paper In Section 3, we give a sequence of three new lemmas which are required for the proof of Theorem 2 In Section 4 we give the proof of Theorem 2 Finally, we prove the three new lemmas in Sections 5, 6 and 7 respectively ) 2 Definitions and assumptions In this section we give umber of definitions and assumptions used in the proof of Theorem 2, which should be taken to be in place throughout the paper First, composing with a rotation if necessary, we may assume that θ 2 0 and that θ = θ 2 With these assumptions, we observe that θ 2 < π/2, and that it follows from (3) that (2) arg( ) θ 2, for n N 0 We choose values of ψ and ψ such that θ 2 < ψ < ψ < π/2, and set σ = cos ψ We define a function (22) µ(r) = M(σr, f), for r > 0 Since it is well-known that (23) log M(r, f) as r,

5 4 D J SIXSMITH we may choose r 0 > 0 sufficiently large that (24) µ(r) > r, for r r 0 and We next define umber of sets First, define, for r > 0, As in [5], we define domains S(r) = {z : arg(z) (ψ θ 2 ), r z }, T (r) = {z : arg(z) (ψ θ 2 ), r z 2r} (25) P κ = {z : Re(z) <, Im(z) 8πκ < 3π}, for κ {, 2, 3} For each a such that f(a) > 0, and for κ {, 2, 3}, we also define a domain (26) Ω κ (a) = {z : Re(z) log f(a) <, Im(z) 8πκ < 3π}, and a compact subset of Ω κ (a) (27) Q κ (a) = {z : 0 Re(z) log f(a) log 2, Im(z) 8πκ ψ θ 2 } We use the notation for a disc B(a, r) = {z : z a < r}, for r > 0 3 Lemmas required for the proof of Theorem 2 In this section we give three preliminary lemmas required for the proof of Theorem 2 These are proved in later sections For ease of comparison we have, where possible, maintained a consistent terminology with that in [5] We first show that, provided that r > 0 is sufficiently large, we have very good control on the size of the modulus of f, and its logarithmic derivative, in T (r) Lemma 3 Suppose that f is a transcendental entire function of the form (2), such that (3) is satisfied, that r 0 is as defined prior to (24), and that µ is the function defined in (22) Then there exists r r 0 such that; (3) (32) (33) (34) f(z) µ( z ), for z S(r ); z f (z) f(z) σ4 4 w f (w) f(w), for z, w T (r), r r ; r f (r) f(r) σ2 log M(r, f) > 0, for r r ; 8 f (z) σ6 64 log M(r, f) f(z), for z T (r), r r r The proof of this lemma is given in Section 5 We note in these equations a significant difference between the properties our family of functions and the properties of the family considered in [5] In [5], a set is constructed in which f(z) is greater than a fixed power of z, and zf (z)/f(z) is comparable to log M( z, f) We next show that, for large values of r > 0, we have that T (r) contains a large number of compact sets each of which is mapped univalently by f onto some T (r ), where r is large compared to r; these are the sets V q,r in the statement of the following lemma Preimages of these sets V q,r are subsequently used to construct a set which lies in A(f) and has Hausdorff dimension 2

6 FUNCTIONS FOR WHICH THE FAST ESCAPING SET HAS DIMENSION TWO 5 If U C is measurable, then we denote the Lebesgue measure of U by area(u) Note that, in the statement of the following lemma, the real valued function t(r) is defined in (6) below, and may be taken to be small compared to r, and the real valued function m(r) is defined in (63) below, and may be taken to be large Lemma 32 Suppose that f is a transcendental entire function of the form (2), such that (3) is satisfied, and that r is as defined in the statement of Lemma 3 Then there exist constants c 2 > 0 and r 2 r with the following property For all r r 2, there exist points b q,r, domains U q,r, and connected compact sets V q,r, such that the following all hold, for q m(r) (i) The discs B(b q,r, t(r)) are pairwise disjoint (ii) f(b q,r ) > 0 (iii) V q,r U q,r B(b q,r, t(r)) B(b q,r, 2t(r)) T (r) (iv) The function f is bounded away from zero in U q,r Moreover, there is a branch of the logarithm and some κ {, 2, 3} such that: (a) U q,r is mapped bijectively by log f onto Ω κ (b q,r ); (b) V q,r is mapped bijectively by log f onto Q κ (b q,r ); (c) V q,r is mapped bijectively by f onto T (f(b q,r )) (v) area(v q,r ) c 2 t(r) 2 The various sets and points constructed in this lemma are illustrated in Figure The proof of this lemma is given in Section 6 t(r) t(r) 3t(r) b q,r b q,r r T(r) 2r Figure Sets and points constructed in Lemma 32 The domain U q,r is shown lightly shaded, and the compact set V q,r is shown shaded darkly In this case, to make the sets easy to identify, we have assumed that m(r) = We use a well-known construction of McMullen in order to estimate the Hausdorff dimension of J(f) A(f) For each n N, suppose that E n is a finite collection of pairwise disjoint compact subsets of C such that the following both hold: (i) If F E n+, then there exists a unique G E n such that F G;

7 6 D J SIXSMITH (ii) If G E n, then there exists at least one F E n+ such that G F We write (35) E n = F E n F, for n N, and E = E n Our final lemma is as follows Here, for measurable sets U and V, we define dens(u, V ) = area(u V ) area(v ) Lemma 33 Suppose that f is a transcendental entire function of the form (2), such that (3) is satisfied, that r 2 is as defined in the statement of Lemma 32 and that µ is the function defined in (22) Then there exists a sequence of finite collections of pairwise disjoint compact sets, (E n ) n N, which satisfies conditions (i) and (ii) above, and, with E and (E n ) n N as defined in (35), such that n N (36) f n (z) S(r 2 ), for n {0,, }, z E In addition there exist a constant c 3 > 0 and a sequence of positive real numbers (d n ) n N, with d n 0 as n, such that if n N and F E n, then (37) dens(e n+, F ) c 3, (38) diam F d n, and n (39) lim n log d n = 0 We prove this lemma in Section 7 4 Proof of Theorem 2 To prove Theorem 2, we require the following three lemmas The first is a key result of McMullen [2, Proposition 22] Lemma 4 Suppose that there exists a sequence of finite collections of pairwise disjoint compact sets, (E n ) n N, which satisfies conditions (i) and (ii) above, and let E and (E n ) n N be as defined in (35) Suppose also that ( n ) n N and (d n ) n N are sequences of positive real numbers, with d n 0 as n, such that for each n N and for each F E n, we have Then dens(e n+, F ) n and diam F d n dim H E 2 lim sup n n k= log k log d n The second, which is a version of [5, Theorem 27], gives an alternative characterisation of A(f) Lemma 42 Suppose that f is a transcendental entire function and that R > 0 is such that µ(r) > r, for r R, where µ is the function defined in (22) Then A(f) = {z : there exists l N such that f n+l (z) µ n (R), for n N} The third lemma is [8, Theorem 3]

8 FUNCTIONS FOR WHICH THE FAST ESCAPING SET HAS DIMENSION TWO 7 Lemma 43 Suppose that f is a transcendental entire function and that z I(f) Set z n = f n (z), for n N Suppose that there exist λ > and N 0 such that (4) f(z n ) 0 and z f (z n ) n f(z n ) λ, for n N Then either z is in a multiply connected Fatou component of f, or z J(f) Now we prove Theorem 2 Let E be the set defined in the statement of Lemma 33 It follows at once from Lemma 33 and Lemma 4 that dim H E = 2 We next show that E A(f) We note that it follows from (3) and (36) that (42) f n (z) µ n ( z ) µ n (r 0 ), for n N, z E The fact that E A(f) follows from (24) and (42), by Lemma 42 It remains to show that either f has a multiply connected Fatou component, or E J(f) It follows from (23) and (34) that there exists R > 0 such that (43) z f (z) f(z) σ6 64 log M(r, f) 2, for z T (r), r R Suppose that z E Since z A(f), there exists N N such that f n (z) R, for n N Hence, by (36) and (43), equation (4) holds with λ = 2 We deduce from Lemma 43 that either z J(f), or z is in a multiply connected Fatou component of f This completes the proof of the theorem 5 Proof of Lemma 3 Recalling that 0 < ψ < π/2 and σ = cos ψ, we observe that, in general, (5) σ ζ Re(ζ), for arg(ζ) ψ We choose { r max r 0, 2 a } N 0 σ, sufficiently large that M(r, f) > and arg(z + ) ψ, for z S(r ), n N We put one additional lower bound on the size of r ; this is after equation (50) It follows from (5), with ζ = z +, that ( ) (52) Re = Re(z + ) σ 2 z + z + 2 Re(z + ), for z S(r ), n N We first prove (3) Let z S(r ), and suppose that w is such that w = σ z and M(σ z, f) = f(w) Then (53) µ( z ) = f(w) c z q + w (54) We claim that n= + w + z, for n N First suppose that n < N 0 Since z r, we have that z w z = ( σ) ( σ) r a 2 N0

9 8 D J SIXSMITH Hence, as required, + w w + z + z On the other hand, suppose that n N 0 It follows, by (5) with ζ = z, that + w ( ) + w = + σ z z + Re + z This completes the proof of (54), and (3) follows from (53) and (54) Next we prove (32) Suppose that r r, and that z, w T (r) Now z f (z) ( f(z) = z q z + ( q ) ( ) ) z + r Re + Re z z + n= It follows from (5), with ζ = z, that ( ) Re = Re(z) z z 2 σ z σ 2r Also, it follows from (52) that ( ) σ 2 Re z + z + Re( ) σ 2, for n N 2r + Re( ) We deduce that (55) z f ( (z) f(z) r qσ 2r + σ2 n= n= n= ) 2r + Re( ) We also have, by (5) with ζ = /(w + ), that w f (w) f(w) q + w w + q + 2r ( ) Re σ w + Now, by (5) with ζ = w, ( ) Re w + It follows that (56) σ 4 4 Re(w + ) rσ + Re( ) 2 σ w f (w) f(w) r ( qσ 4 4r + σ2 n= n=, for n N 2r + Re( ) ) 2r + Re( ) Equation (32) follows from (55) and (56) Next we prove (33) Suppose that r r It follows from (52), with z = r, that (57) r f (r) f(r) = q + r (r + ) q + ( ) r Re q + σ 2 r r + r + n= Suppose that w is such that w = r and M(r, f) = f(w) Then log M(r, f) log f(w) log c (58) = + q + n= log ( + r/ ) n= Let N N be such that, for n N We claim that (59) r r + 4 log ( + r/ ), for n N n=

10 FUNCTIONS FOR WHICH THE FAST ESCAPING SET HAS DIMENSION TWO 9 To prove this claim, suppose that n N We consider first the case that 2 < r In this case r r + > 2 = log r2 2 4 log ( + r/ ) This completes the proof of the claim in this case We consider next the case that 2 r It is readily seen by differentiation that the function ( h(x) = ( + x) log + ), for x > 0, x is decreasing We deduce that 3 ( 2 log 3 ( + x) log + ), for x x 2 It follows that ( + a ) ( n log + r ) r This completes the proof of our claim (59) By (59), we deduce from (57) and (58) that ( r f (r) f(r) σ2 log M(r, f) σ2 log( + r/ ) 8 8 n=n ( σ2 (50) 8 log( + r/ ) n=n N log c n= log c N ) log( + r/ ) ) log( + r/ a ) The first term on the right-hand side of (50) tends to infinity, as r tends to infinity; however, the other terms are bounded Hence, for a sufficiently large choice of r, this is sufficient to establish (33) Finally (34) follows from (32) with w = r, and from (33) 6 Proof of Lemma 32 We use the following version of the Ahlfors five islands theorem; see, for example, [0, Theorem 62] Lemma 6 Suppose that D κ, κ {, 2, 3}, are Jordan domains with pairwise disjoint closures Then there exists ν > 0 with the following property If r > 0 and f : B(a, r) C is an analytic function such that f (a) + f(a) 2 ν r, then there exists κ {, 2, 3} such that B(a, r) has a subdomain which is mapped bijectively onto D κ To prove Lemma 32 we first fix a value of ν > 0 such that the conclusion of Lemma 6 is satisfied for the domains defined in (25) Since, by (33), f (r) 0, for r r, we may define a function (6) t(r) = 8ν σ 4 f(r) f (r), for r r

11 0 D J SIXSMITH We observe by (33) that (62) t(r) 64ν σ 6 log M(r, f) r, for r r It follows from (23) and (62) that we may assume that t(r) is small compared to r, provided that r is sufficiently large We deduce that there exist r 2 r and c > 0 such that if r r 2, then T (r) contains at least (63) m(r) = c r 2 t(r) 2 disjoint discs of radius 3t(r) We may assume that m(r) is large, for r r 2 Suppose that r r 2 We choose a sequence of points (β q,r ), for q m(r), such that the discs B(β q,r, 3t(r)) are pairwise disjoint, and such that B(β q,r, 3t(r)) T (r), for q m(r) Choose one of the points (β q,r ), which we denote by β We also write t for t(r) We construct a point b, a domain U, and a compact set V such that, with b q,r = b, U q,r = U and V q,r = V, the conclusions of the lemma are satisfied First we note that f(z) 0, for z B(β, 3t), by (3), and so we may define a branch of log f in this disc Let h β : B(β, t) C be defined by h β (z) = log f(z) log f(β) We have h β (β) = 0 Hence, by (32) with z = β and w = r, by (6), and since β 2r, h β (β) + h β (β) 2 = h β(β) = f (β) f(β) σ4 8 f (r) f(r) = ν t Hence, by Lemma 6, there is a subdomain of B(β, t) which is mapped bijectively by h β onto P κ, for some κ {, 2, 3}, where P κ is as defined in (25) In particular, there exists b B(β, t) such that f(b) > 0 We next let h b : B(b, t) C be defined by h b (z) = log f(z) log f(b) We have h b (b) = 0 and so, as above, h b (b) + h b (b) 2 ν t Applying Lemma 6 a second time, there is a subdomain U of B(b, t) which is mapped bijectively by h b onto P κ, for some κ {, 2, 3} It follows that log f maps U bijectively onto Ω κ (b) Let V be the subset of U mapped bijectively by log f onto Q κ (b) It follows that V is mapped bijectively by f onto T (f(b)) This establishes part (iv) of the lemma We note that parts (i) and (ii) of the lemma are immediate, and the set inclusions in part (iii) of the lemma hold by construction Finally, we need to estimate the area of V Since Q κ (b) = log f(v ), we deduce by (32) with z replaced by r, by (6), and since w r, that area(q κ (b)) = 2(ψ θ 2 ) log 2 sup f 2 (w) area(v ) w V ( 4 σ 4 f(w) f (r) 2 f(r) ) area(v ) = 024ν2 σ 6 area(v ) t2

12 FUNCTIONS FOR WHICH THE FAST ESCAPING SET HAS DIMENSION TWO Hence area(v ) c 2 t 2, where c 2 = (ψ θ 2)σ 6 log 2 52ν 2 7 Proof of Lemma 33 In this section we construct the sequence of finite collections of sets (E n ) n N to which Lemma 4 is applied Our construction is very similar to that of Bergweiler and Karpińska in [5] The main difference, apart from minor changes in notation, is that their construction is within a whole annulus whereas ours is within a closed sector of an annulus T (r), for r r 2 ; this makes some parts of our construction slightly easier We require the following [5, Lemma 4] Lemma 7 Suppose that Ω is a domain, and that Q Ω is compact Then there exists C = C(Ω, Q) > 0 such that if g is analytic and univalent in Ω, then g (w) C g (z), for w, z Q In particular, recalling the definitions given in (26) and (27), we deduce the following Lemma 72 There exists C > 0 such that if κ {, 2, 3}, a is such that f(a) > 0 and g is analytic and univalent in Ω κ (a), then g (w) C g (z), for w, z Q κ (a) Proof This follows from Lemma 7, since the sets Ω κ (a) and Q κ (a) are all translations of fixed sets We also, for convenience, use the following version of the Koebe distortion theorem; see, for example, [5, Lemma 42] Lemma 73 Suppose that r > 0 and that g is analytic and univalent in the disc B(a, r) Then We require one other result g (z) 2 g (a), for z B(a, r/2) Lemma 74 Suppose that f is a transcendental entire function Then there exists R 0 = R 0 (f) > 0 such that log log M n (R 0, f) n, as n Proof This result follows immediately from the fact noted in [4, Equation (6)] that, for sufficiently large values of R, log log M(R, f n ) n, as n We now prove Lemma 33 We first recall the various sets and points constructed in the proof of Lemma 32, in particular, for r r 2, the points b q,r, domains U q,r and connected compact sets V q,r, for q m(r)

13 2 D J SIXSMITH It is well-known (see, for example, [5]) that if k >, then M(kr, f) M(r, f) as r It follows that we may choose R R 0, where R 0 is as defined in the statement of Lemma 74, sufficiently large that { } (7) M(r, f) max 2, σ 2 M(σr, f), for r R Choose ρ 0 > 0 sufficiently large that σ 2 ρ 0 max{r 2, R } We claim that { } (72) µ k (ρ 0 ) max 2, σ 2 M k (σ 2 ρ 0 ) 2M k (R ), for k N The right-hand inequality of (72) is obvious The left-hand inequality of (72) is obtained by induction It is clearly true for k =, and when k = n + we have ( ) µ n+ (ρ 0 ) µ σ 2 M n (σ 2 ρ 0 ) inductively by (72) { } max 2, σ 2 M n+ (σ 2 ρ 0 ) by (24) and (7) Put E 0 = {T (ρ 0 )} and E = {V q,ρ0 : q m(ρ 0 )} We claim that we may define collections of compact sets (E n ) n {0,,} such that if F E n+, then the following hold The function f n+ : F T (ρ n+,f ), for some ρ n+,f > ρ 0, is a bijection If G E n is such that F G, then there exists q {,, m(ρ n,g )} such that f n (F ) = V q,ρn,g T (ρ n,g ) The sequence (E n ) n N has the properties (i) and (ii) discussed before the statement of Lemma 4 We establish this claim inductively First we note that if n = 0, then the claim follows straightforwardly from Lemma 32 Suppose then that n N and that G E n For simplicity we write ρ for ρ n,g See Figure 2 for a simple picture of the sets used in this proof Since, by assumption, f n : G T (ρ) is a bijection, we may let h be the branch of the inverse of f n which maps T (ρ) to G We then set E n+ (G) = {h(v q,ρ ) : q m(ρ)} Finally we set E n+ = E n+ (G) G E n The claimed properties now follow by construction, and by Lemma 32 Note that (36) also follows from the construction Fix F E n, for some n N, and set ρ = ρ n,f for simplicity We need to establish (37), which concerns densities of sets, and (38) and (39), which concern the diameters of sets First we prove (37) We deduce from Lemma 32 that T (ρ) = exp W, where W = Q κ (a) for some κ {, 2, 3} and some a such that f(a) = ρ Moreover, the function log f n : F W is a bijection, and the inverse branch of log f n, h : W F, extends to Ω κ (a)

14 FUNCTIONS FOR WHICH THE FAST ESCAPING SET HAS DIMENSION TWO 3 ρ T(ρ) 2ρ Figure 2 f n (G) = T (ρ), where G E n, containing m(ρ) balls of radius t(ρ), each of which contains one of the V q,ρ, shown shaded Note that, by construction, each ball is a distance of at least t(ρ) from the boundary of T (ρ) The preimages under f n of the V q,ρ make up the sets E n+ (G) Suppose that q m(ρ) Let W q W be such that exp W q = V q,ρ Since z 2ρ, for z V q,ρ, we deduce that area(w q ) = dx dy V q,ρ z 2 4ρ 2 area(v q,ρ) It follows by Lemma 32, and by (63), that m(ρ) area W q c 2t(ρ) 2 4ρ 2 m(ρ) = c c 2 4 q= Hence, by Lemma 72, m(ρ) dens(e n+, F ) = dens h(w q ), h(w ) q= m(ρ) C 2 dens W q, W q= c c 2 4C 2 area(w ) c c 2 = 8C 2 (ψ θ 2 ) log 2, where C is the constant in Lemma 72 This completes the proof of (37), with c c 2 c 3 = 8C 2 (ψ θ 2 ) log 2

15 4 D J SIXSMITH Finally we prove (38) and (39) We may assume that n 2 Suppose that F m E m is such that F F m, for 0 m < n For simplicity we write ρ m for ρ m,fm It follows from Lemma 32 that f n maps F n bijectively to T (ρ n ), and that f n (F ) = V q,ρn B(b q, t(ρ n )) B(b q, 2t(ρ n )) T (ρ n ), for some q {,, m(ρ n )} We let h be the branch of the inverse of f n which maps f n (F ) to F, and note that h may be extended to a function which is univalent in B(b q, 2t(ρ n )) Hence, by Lemma 73, we have that It follows that h (z) 2 h (b q ), for z B(b q, t(ρ n )) (73) diam F 2 h (b q ) diam f n (F ) 24 h (b q ) t(ρ n ) Now, let ζ = h(b q ) It is well-known that log M(r, f) is an increasing function of r Since f m (ζ) T (ρ m ), for 0 m < n, it follows from (34) that f (f m (ζ)) c 4 log M(ρ m, f) log ρ m ρ m+ ρ m, for 0 m < n, where c 4 = σ 6 /64 It follows by the chain rule that (74) h (b q ) ρ n 2 0 log ρ m ρ n c m=0 4 log M(ρ m, f) (75) Now, by (33) and (6), log M(ρ n, f) log ρ n 8ρ n σ 2 We deduce from (73), (74) and (75) that diam F 24ρ n 2 0 ρ n m=0 f (ρ n ) f(ρ n ) = 64νρ n σ 6 t(ρ n ) log ρ m c 4 log M(ρ m, f) t(ρ n ) c n 5 c n 4 m=0 log ρ m log M(ρ m, f), where c 5 = 536νρ 0 /σ 6 We recall the definition of the function µ from (22), and note that, by (3), (76) 2ρ m f m (ζ) µ m (ρ 0 ), for 0 m < n It follows from (72) and (76) that ρ m M m (R ) for 0 m < n, where R is as in (7) We deduce that log M(ρ m, f) log M m+ (R ) log ρ m log M m, for 0 m < n (R ) It follows that diam F c 5 c n log R 4 log M n (R ) = exp (c 6 n log c 4 log log M n (R )),

16 FUNCTIONS FOR WHICH THE FAST ESCAPING SET HAS DIMENSION TWO 5 where c 6 = log (c 4 c 5 log R ) We set d n = exp (c 6 n log c 4 log log M n (R )), for n N By Lemma 74, n lim n log d n = lim n n n log c 4 + log log M n (R ) c 6 = 0, and so (39) is satisfied It also follows that d n 0 as n This completes the proof of the lemma Acknowledgment: The author is grateful to Gwyneth Stallard and Phil Rippon for all their help with this paper, and to the referee for helpful suggestions References [] Abramowitz, M, and Stegun, I A, Eds Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables National Bureau of Standards, 972 [2] Ahlfors, L V Complex analysis: An introduction of the theory of analytic functions of one complex variable Second edition McGraw-Hill Book Co, New York, 966 [3] Bergweiler, W Iteration of meromorphic functions Bull Amer Math Soc (NS) 29, 2 (993), 5 88 [4] Bergweiler, W, and Hinkkanen, A On semiconjugation of entire functions Math Proc Cambridge Philos Soc 26, 3 (999), [5] Bergweiler, W, and Karpińska, B On the Hausdorff dimension of the Julia set of a regularly growing entire function Math Proc Cambridge Philos Soc 48, 3 (200), [6] Bergweiler, W, Rippon, P J, and Stallard, G M Multiply connected wandering domains of entire functions Proc London Math Soc 07, 6 (203), [7] Eremenko, A E On the iteration of entire functions Dynamical systems and ergodic theory (Warsaw 986) 23 (989), [8] Eremenko, A E, and Lyubich, M Y Dynamical properties of some classes of entire functions Ann Inst Fourier (Grenoble) 42, 4 (992), [9] Falconer, K Fractal Geometry: Mathematical Foundations and Applications, second ed Wiley, 2006 [0] Hayman, W K Meromorphic functions Oxford Mathematical Monographs Clarendon Press, Oxford, 964 [] Langley, J K On the multiple points of certain meromorphic functions Proc Amer Math Soc 23, 6 (995), [2] McMullen, C Area and Hausdorff dimension of Julia sets of entire functions Trans Amer Math Soc 300, (987), [3] Rippon, P J, and Stallard, G M Dimensions of Julia sets of meromorphic functions J London Math Soc (2) 7, 3 (2005), [4] Rippon, P J, and Stallard, G M On questions of Fatou and Eremenko Proc Amer Math Soc 33, 4 (2005), 9 26 [5] Rippon, P J, and Stallard, G M Fast escaping points of entire functions Proc London Math Soc (3) 05, 4 (202), [6] Rippon, P J, and Stallard, G M Baker s conjecture and Eremenko s conjecture for functions with negative real zeros J Anal Math 20, (203), [7] Sixsmith, D J Simply connected fast escaping Fatou components Pure Appl Math Q 8, 4 (202), [8] Sixsmith, D J Julia and escaping set spiders webs of positive area Preprint, arxiv: v (203) Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK address: davidsixsmith@openacuk