ON THE DIFFERENTIABILITY OF HAIRS FOR ZORICH MAPS PATRICK COMDU¨HR Abstract. Devaney and Krych showed that for the exponential family λez, 7 where0<λ<1/e,theJuliasetconsistsofuncountablymanypairwisedisjoint 1 simple curves tending to ∞. Viana proved that these curves are smooth. In 0 this article we consider a quasiregular counterpart of the exponential map, the 2 so-called Zorich maps, and generalize Viana’s result to these maps. n a J 4 2 1. Introduction and main result ] For an entire function f the Julia set J(f) of f is the set of all points in C where S D the iterates fk of f do not form a normal family in the sense of Montel. Given . an attracting fixed point ξ of f we denote by A(ξ) := {z : lim fk(z) = ξ} h k→∞ t the basin of attraction of ξ. From the theory of complex dynamics it is well- a m known that J(f) = ∂A(ξ), see [Mi06, Corollary 4.12]. For further information on complex dynamics we refer to [Bea91, Ber93, Mi06, St93]. [ Devaney and Krych [DK84] showed that for f(z) = λez, where 0 < λ < 1/e, 1 there exists an attracting fixed point ξ ∈ R such that J(f) = C\A(ξ) and gave v 3 a detailed description of the structure of J(f). We say that a subset H of C 7 (or Rd) is a hair, if there exists a homeomorphism γ: [0,∞) → H such that 8 6 lim γ(t) = ∞. We call γ(0) the endpoint of the hair. t→∞ 0 We only state the part of the result due to Devaney and Krych which is relevant . 1 for us. 0 7 Theorem A. For 0 < λ < 1/e the set J(λez) is an uncountable union of pairwise 1 : disjoint hairs. v Xi For a set X in C (or in Rd) we denote by dimX the Hausdorff dimension of r X. The following result is due to McMullen [McM87]. a Theorem B. Let λ ∈ C\{0}. Then dimJ(λez) = 2. McMullen’s result implies that in the situation of Theorem A the union of the hairs has Hausdorff dimension 2. The following result of Karpin´ska [Ka99] is also known as Karpin´ska’s paradox, see e.g. [SZ03]. Theorem C. Let 0 < λ < 1/e and let C be the set of endpoints of the hairs that form J(λez). Then dimC = 2 and dim(J(λez)\C) = 1. 2010 Mathematics Subject Classification. 37F10 (primary), 30C65, 30D05 (secondary). Keywordsandphrases. Exponentialmap,Zorichmap,quasiregularmap,complexdynamics, hair, external ray. 1 2 PATRICK COMDU¨HR The existence of hairs is not restricted to the situation considered by Devaney and Krych. Hairs appear for λez for all λ ∈ C\{0} (see [DGH86, SZ03]) and also for more general classes of functions (see [Ba07, DT86, RRRS11]). For further information on dynamics of exponential functions we refer to papers by Rempe [Re06] and Schleicher [Sch03]. Viana [Vi88] investigated the differentiability of hairs for exponential maps. Theorem D. For all λ ∈ C\{0} the hairs of λez are C∞-smooth. In this paper we consider a higher-dimensional analog of exponential maps, the so-called Zorich maps. These maps are quasiregular maps, which can be considered as a higher-dimensional analog of holomorphic maps. Since we will not use any results about quasiregular maps, we do not give the definition here, but refer to Rickman’s monograph [Ri93]. We note, however, that the quasiregularity is an underlying idea in many of the arguments. Every Zorich map [Zo67] depends on a bi-Lipschitz map which maps a square to the upper or lower hemisphere, so there are plenty ways to define such a map. For the definition of these maps we follow [IM01, page 119]. Subsequently we summarize some results of Bergweiler [Ber10], but we replace the dimension 3 by any dimension d ≥ 3. We define for d ∈ N with d ≥ 3 the hypercube Q := {x ∈ Rd−1 : (cid:107)x(cid:107) ≤ 1} = [−1,1]d−1, ∞ the upper hemisphere S := {x ∈ Rd : (cid:107)x(cid:107) = 1 and x ≥ 0} + 2 d and for c ∈ R the half-space H := {x ∈ Rd : x ≥ c}. ≥c d The half-spaces H , H and H are defined analogously. For a bi-Lipschitz >c <c ≤c map h: Q → S we define + F: Q×R → H , F(x) = exdh(x ,...,x ). ≥0 1 d−1 By reflection we get a function F: Rd → Rd which we call a Zorich map. If DF(x ,...,x ,0) exists, we have 1 d−1 (1.1) DF(x ,...,x ,x ) = exdDF(x ,...,x ,0) 1 d−1 d 1 d−1 and thus there exist α,m,M ∈ R, α ∈ (0,1), m < M, M ≥ 1 such that (1.2) (cid:107)DF(x)(cid:107) ≤ α a.e. for x ≤ m d and 1 (1.3) l(DF(x)) ≥ a.e. for x ≥ M, d α where DF(x) denotes the derivative, (cid:107)DF(x)(cid:107) = sup (cid:107)DF(x)h(cid:107) 2 (cid:107)h(cid:107) =1 2 ON THE DIFFERENTIABILITY OF HAIRS FOR ZORICH MAPS 3 the operator norm of DF(x) and l(DF(x)) = inf (cid:107)DF(x)h(cid:107) . 2 (cid:107)h(cid:107) =1 2 Consider now for a ∈ R such that (1.4) a ≥ eM −m the map f : Rd → Rd, f (x) = F(x)−(0,...,0,a). a a In our context we call f a Zorich map, too. The following theorem is due to a Bergweiler [Ber10]. Theorem E. Let f be as above with a as in (1.4). Then there exists a unique a fixed point ξ = (ξ ,...,ξ ) satisfying ξ ≤ m, the set 1 d d J := {x ∈ Rd: fk(x) (cid:57) ξ} a consists of uncountably many pairwise disjoint hairs, and the set C of endpoints of these hairs has Hausdorff dimension d, while J \C has Hausdorff dimension 1. Thus Theorem E corresponds to Theorem A, B and C. For a set Ω ⊂ Rd−1 we denote by int(Ω) the interior of Ω. The main result of this article is the following. Theorem 1.1. Let f be as above with a as in (1.4) and assume that h| is a int(Q) C1 and Dh is H¨older continuous. Then the hairs of f are C1-smooth. a This theorem corresponds to Theorem D concerning the differentiability of hairs. As in Viana’s result, we do not obtain differentiability of the hairs in the endpoints since they can spiral around a point. Rempe [Re03, 3.4.2 Theorem, page 31] gave a condition under which we obtain smoothness up to the endpoints for exponential maps. The proof of Theorem 1.1 will show that the conclusion of the theorem still holds if the assumptions on h and Dh are satisfied on a suitable subset of int(Q) (see Section 4). Acknowledgement. I would like to thank my supervisor Walter Bergweiler for valuable suggestions and constant support. 2. Preliminaries In this section we are recalling mainly results from [Ber10], formulating them for functions in Rd with d ≥ 3 instead of R3, however. For simplicity we write f = (f ,...,f ) instead of f . For r ∈ Zd−1 we denote by 1 d a P(r) := P(r ,...,r ) := {x ∈ Rd−1: ∀j ∈ {1,...,d−1}: |x −2r | < 1} 1 d−1 j j = int(Q)+(2r ,...,2r ) 1 d−1 the shifted and open square Q with centre 2r. We put (cid:40) (cid:41) d−1 (cid:88) S := r ∈ Zd−1: r ∈ 2Z . j j=1 4 PATRICK COMDU¨HR Then (cid:26)H , if r ∈ S F maps P(r)×R bijectively onto >0 H , if r ∈/ S <0 and thus (cid:26)H , if r ∈ S, f maps P(r)×R bijectively onto >−a H , if r ∈/ S. <−a Definition 2.1 (Tract). For r ∈ S we call the set T(r) := P(r)×(M,∞) the tract above P(r). Now we want to understand the behaviour of our function f and collect some important facts about it. Since f(P(r)×R) = H for r ∈ S and >−a (2.1) f (x ,...,x ) = exdh (x ,...,x )−a ≤ exd −eM +m ≤ m < M d 1 d d 1 d−1 for x ≤ M and hence f(P(r) × (−∞,M]) ⊂ H , we have f(T(r)) ⊃ H . d <M ≥M Thus there exists a branch Λr: H → T(r) of the inverse function of f. Using ≥M the notation Λ := Λ(0,...,0), we have (2.2) Λ(r1,...,rd−1)(x) = Λ(x)+(2r ,...,2r ,0) 1 d−1 for all x ∈ H and r ∈ S. ≥M In our proofs we will often use the derivative of Λr. Together with (2.2) we obtain DΛr(x) = DΛ(x) for all x ∈ H for which the derivative exists. Because DF = Df, we deduce ≥M from (1.2) that (2.3) (cid:107)DΛ(x)(cid:107) ≤ α a.e. for x ∈ H . This implies for x,y ∈ H that ≥M ≥M (2.4) (cid:107)Λ(x)−Λ(y)(cid:107) ≤ (cid:107)x−y(cid:107) esssup(cid:107)DΛ(z)(cid:107) ≤ α(cid:107)x−y(cid:107) . 2 2 2 z∈[x,y] Since (2.1) and thus f(H ) ⊂ H , we obtain using (1.3) ≤M ≤m (cid:107)f(x)−f(y)(cid:107) ≤ (cid:107)x−y(cid:107) esssup(cid:107)Df(z)(cid:107) ≤ α(cid:107)x−y(cid:107) . 2 2 2 z∈[x,y] So Banach’s fixed point theorem gives us the existence of a unique fixed point ξ ∈ H such that lim fn(x) = ξ for all x ∈ H . This yields ≤m ≤m n→∞ (cid:91) J ⊂ T(r). r∈S In the following we collect some estimates for DΛ. The equation (1.1) implies that there exist c ,c > 0 such that 1 2 c exd ≤ l(Df(x)) ≤ (cid:107)Df(x)(cid:107) ≤ c exd a.e. 1 2 ON THE DIFFERENTIABILITY OF HAIRS FOR ZORICH MAPS 5 Thus there are constants c ,c > 0 such that for x ∈ H 3 4 ≥M c 3 l(DΛ(x)) ≥ a.e. (cid:107)x(cid:107) 2 c 4 (2.5) (cid:107)DΛ(x)(cid:107) ≤ a.e. (cid:107)x(cid:107) 2 and since Λ is quasiregular, there exist c ,c > 0 such that 5 6 c c 5 6 ≤ J (x) ≤ a.e., (cid:107)x(cid:107)d Λ (cid:107)x(cid:107)d 2 2 where J (x) denotes the Jacobian determinant. Λ Let us fix x,y ∈ H . Then we can connect x and y by a path γ in ≥M (cid:110) (cid:111) H ∩ z ∈ Rd: (cid:107)z(cid:107) ≥ min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) } ≥M 2 2 2 with length(γ) ≤ π(cid:107)x−y(cid:107) . Together with (2.5) this yields 2 (cid:107)x−y(cid:107) (2.6) (cid:107)Λ(x)−Λ(y)(cid:107) ≤ π(cid:107)x−y(cid:107) esssup(cid:107)DΛ(z)(cid:107) ≤ c π 2 . 2 2 4 min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) } z∈γ 2 2 Since h is bijective, there exists a unique point (v ,...,v ) ∈ Q which is mapped 1 d−1 under h to the north pole of the sphere, i.e. (2.7) h(v ,...,v ) = (0,...,0,1). 1 d−1 This implies F(v ,...,v ,x ) = (0,...,0,exd) and for r ∈ S and t ≥ M this 1 d−1 d yields (cid:0) (cid:1) Λr(0,...,0,t) = v +2r ,...,v +2r ,log(t+a) . 1 1 d−1 d−1 Moreover, the equation (cid:107)F(x)(cid:107) = exd 2 yields (cid:107)f(Λ(x))+(0,...,0,a)(cid:107) = eΛd(x) 2 and thus (2.8) Λ (x) = log((cid:107)x+(0,...,0,a)(cid:107) ). d 2 To discuss the existence of hairs, it is useful, following Schleicher and Zimmer [SZ03], to define a reference function E: [0,∞) → [0,∞), E(t) = et −1. This function has the following properties: 1. We have E(0) = 0 and lim Ek(t) = ∞ for t > 0. k→∞ 2. For b > 1 we have (cid:0) (cid:1) (2.9) Ek(t) < log Ek+1(t)+b ≤ Ek(t)+log(b). 3. If 0 < t(cid:48) < t(cid:48)(cid:48) < ∞, then Ek(t(cid:48)(cid:48)) (cid:0) (cid:1) (2.10) lim Ek(t(cid:48)(cid:48))−Ek(t(cid:48)) = ∞ and lim = ∞. k→∞ k→∞ Ek(t(cid:48)) 6 PATRICK COMDU¨HR We need an analogous definition as in the case of exponential maps: Definition 2.2 (External address/Symbolic space). For each x ∈ J we call the sequence N s(x) := s s s ··· = (s ) ∈ S 0 0 1 2 k k≥0 such that fk(x) ∈ T(s ) for all k ≥ 0 the external address of x. Moreover, we call k Σ := SN0 the symbolic space. Definition 2.3 (Admissibility/exponentially boundedness). We say that s ∈ Σ is admissible (or exponentially bounded), if there exists a t > 0 such that (cid:107)s (cid:107) limsup k 2 < ∞. Ek(t) k→∞ Moreover, we denote by Σ(cid:48) ⊂ Σ the set of all admissible points. With these definitions we obtain the following lemmas, see [Ber10, Propositions 3.1 and 3.2]. Lemma 2.4. Let x ∈ J. Then s(x) is admissible. Lemma 2.5. Let s ∈ Σ(cid:48). Then {x ∈ J : s(x) = s} is a hair. From Lemma 2.4 and Lemma 2.5 it follows that J is the union of hairs as stated in Theorem E. Fixing s ∈ Σ(cid:48), we denote (cid:26) (cid:27) (cid:107)s (cid:107) t := inf t > 0: limsup k 2 < ∞ . s Ek(t) k→∞ Choosing t ∈ [0,∞) such that 2(cid:107)s (cid:107) = Ek(t ) and putting τ := supt , we k k 2 k k j j≥k obtain (2.11) t = limsupt = lim τ . s k k k→∞ k→∞ Using the abbreviation L := Λsk = Λ(sk,1,...,sk,d−1) k we define for k ∈ N 0 (2.12) g : [0,∞) → H , g (t) = (L ◦L ◦···◦L )(cid:0)0,...,0,Ek+1(t)+M(cid:1). k ≥M k 0 1 k ThetwomainlemmasintheproofofLemma2.5arethefollowing[Ber10, Lemmas 3.1 and 3.2]. Lemma 2.6. The sequence (g ) converges locally uniformly on (t ,∞). k k≥0 s Lemma 2.7. The sequence (g ) has a subsequence which converges uniformly k k≥0 on [t ,∞) and thus g extends to a continuous map g: [t ,∞) → H . s s ≥M ON THE DIFFERENTIABILITY OF HAIRS FOR ZORICH MAPS 7 3. Proof of Theorem 1 To have a chance for a C1 condition for our hairs, we need enough regularity of the bi-Lipschitz mapping h. In this section we want to specify this condition and want to give precise C1 estimates for the hairs. Therefore we will introduce some new notations. For n,m ∈ N with m < n and functions f ,...,f : Rd → Rd we denote 1 n n (cid:13) f := f ◦···◦f , j m n j=m where we use the convention m (cid:13) f := id. j j=m+1 Then g defined by (2.12) takes the form k (cid:18) (cid:19) k (cid:0) (cid:1) g (t) = (cid:13) L 0,...,0,Ek+1(t)+M k j j=0 for all k ∈ N and t ∈ [0,∞). If h is locally C1, the derivative g(cid:48) then reads as 0 k d (cid:0) (cid:0) (cid:1)(cid:1) g(cid:48)(t) = (L ◦···◦L ) 0,...,0,Ek+1(t)+M k dt 0 k k+1(cid:18) (cid:18)(cid:18) (cid:19) (cid:19)(cid:19) (3.1) = (cid:89) DΛ (cid:13)k L (0,...,0,Ek+1(t)+M) j j=l l=1 ·(cid:0)0,...,0,(Ek+1)(cid:48)(t)(cid:1)T for k ∈ N. To prove Theorem 1.1, it is enough to show the following result. Theorem 3.1. Let f be as before and assume that h| is C1 and Dh is H¨older int(Q) continuous. Then for all x ∈ J the sequence (g ) consists of C1-curves which k k≥0 converge (in C1-sense) locally uniformly on (t ,∞). s For the proof of the theorem we will compare g(cid:48) and g(cid:48) in a suitable way. k k−1 Therefore we define as a preparation for all k ∈ N the auxilary function 0 (3.2) φ : [0,∞) → H , φ (t) = L (0,...,0,Ek+1(t)+M). k ≥M k k Then we obtain for all t ≥ 0 (cid:0) (cid:1) φ (t) = v +2s ,...,v +2s ,log(Ek+1(t)+M +a) . k 1 k,1 d−1 k,d−1 Thus (cid:18) (cid:19) 1 (3.3) φ(cid:48)(t) = 0,...,0, E(cid:48)(Ek(t))·(Ek)(cid:48)(t) k Ek+1(t)+M +a and (3.4) φ(cid:48)(t) = DΛ(cid:0)0,...,0,Ek+1(t)+M(cid:1)·(cid:16)0,...,0,(cid:0)Ek+1(cid:1)(cid:48)(t)(cid:17)T . k Noticing that (3.5) a ≥ eM −m ≥ 1+M −m > 1, 8 PATRICK COMDU¨HR we obtain Ek+1(t)+1 (3.6) (cid:107)φ(cid:48)(t)(cid:107) = ·(Ek)(cid:48)(t) ≤ (Ek)(cid:48)(t). k 2 Ek+1(t)+M +a Lemma 3.1. For all k ∈ N and l ∈ {1,...,k −1} and for all t ≥ 0 we have ≥2 (cid:13)(cid:18) (cid:19) (cid:13) (cid:18) (cid:19) (cid:13) k−1 (cid:13) k−1 (3.7) (cid:13) (cid:13) L (φ (t))(cid:13) ≥ (cid:13) L (φ (t)) ≥ El(t). j k j k (cid:13) (cid:13) j=l j=l 2 d Proof. From (2.8), (2.9) and (3.5) we deduce that (cid:107)L (φ (t))(cid:107) ≥ |L (φ (t))| k−1 k 2 k−1,d k = |Λ (φ (t))| d k = log((cid:107)φ (t)+(0,...,0,a)(cid:107) ) k 2 ≥ log(φ (t)+a) k,d (cid:0) (cid:0) (cid:1) (cid:1) = log log Ek+1(t)+M +a +a ≥ log(Ek(t)+a) ≥ Ek−1(t) Take now l < k−1 such that (3.7) is true with l replaced by l+1. Then we obtain (cid:13)(cid:18) (cid:19) (cid:13) (cid:13) (cid:18) (cid:19) (cid:13) (cid:13) k−1 (cid:13) (cid:13) k−1 (cid:13) (cid:13) (cid:13) L (φ (t))(cid:13) = (cid:13)L ◦ (cid:13) L (φ (t))(cid:13) j k l j k (cid:13) (cid:13) (cid:13) (cid:13) j=l j=l+1 2 2 (cid:12) (cid:18)(cid:18) (cid:19) (cid:19)(cid:12) (cid:12) k−1 (cid:12) ≥ (cid:12)L (cid:13) L (φ (t)) (cid:12) l,d j k (cid:12) (cid:12) j=l+1 (cid:18)(cid:18) (cid:19) (cid:19) k−1 = log (cid:13) L (φ (t))+(0,...,0,a) j k j=l+1 (cid:32) (cid:33) (cid:18) (cid:19) k−1 ≥ log (cid:13) L (φ (t))+a j k j=l+1 d ≥ log(El+1(t)+a) ≥ El(t) which proves the result. (cid:3) Remark 1. The argument shows that the conclusion of Lemma 3.1 also holds if φ (t) is replaced by (0,...,0,Ek(t)+M). k Since the operatornorm of DF(x) is comparable to the maximum of all entries of this matrix, there exists a constant C > 0 such that (3.8) (cid:107)DF(x)(cid:107) ≤ C max |DF (x)|. jk 1≤j≤d 1≤k≤d In the following let β ∈ (0,1] and let Dh be β-Ho¨lder continuous, i.e. there is a constant H > 0 such that β (cid:107)Dh(x)−Dh(y)(cid:107) ≤ H (cid:107)x−y(cid:107)β β 2 ON THE DIFFERENTIABILITY OF HAIRS FOR ZORICH MAPS 9 for all x,y ∈ H . Moreover, we denote by L the Lipschitz constant of h. ≥M h We want to use this to prove the following Lemma: Lemma 3.2. If h is as in Theorem 3.1, there is a constant c > 0 such that for 7 all x,y ∈ H ≥M (cid:110) (cid:111) (cid:110) (cid:111) (cid:107)DF(Λ(x))−DF(Λ(y))(cid:107) ≤ c min (cid:107)x(cid:107)1−β,(cid:107)y(cid:107)1−β ·max (cid:107)x−y(cid:107) ,(cid:107)x−y(cid:107)β . 7 2 2 2 2 Proof. We have for x = (x˜,x ),y = (y˜,y ) ∈ H using (3.8) d d ≥M (cid:107)DF(x)−DF(y)(cid:107) ≤ C max |DF (x)−DF (y)| jk jk 1≤j≤d 1≤k≤d ≤ C max |exdDF (x˜,0)−eydDF (y˜,0)|. jk jk 1≤j≤d 1≤k≤d Because |DF (x)−DF (y)| is symmetric in x and y, we assume without loss of jk jk generality that y ≤ x . Then we obtain d d |exdDF (x˜,0)−eydDF (y˜,0)| jk jk ≤ eyd|DF (x˜,0)−DF (y˜,0)|+(exd −eyd)·max{|DF (x˜,0)|,|DF (y˜,0)|}. jk jk jk jk Since h is as in Theorem 3.1 and for all z ∈ Rd and j,k ∈ {1,...,d}  ∂  h (z˜), if k ≤ d−1, j DF (z˜,0) = ∂x jk k  h (z˜), if k = d, j (cid:12) (cid:12) we have with C˜ = max{L ,H } ≥ 1, noting that (cid:12) ∂ h (z˜)(cid:12) ≤ L , h β (cid:12)∂x j (cid:12) h k (cid:16) (cid:110) (cid:111) (cid:17) (cid:107)DF(x)−DF(y)(cid:107) ≤ C C˜eyd ·max (cid:107)x˜−y˜(cid:107) ,(cid:107)x˜−y˜(cid:107)β +L ·(exd −eyd) 2 2 h (cid:16) (cid:110) (cid:111) (cid:17) ≤ CC˜ eyd ·max (cid:107)x−y(cid:107) ,(cid:107)x−y(cid:107)β +(exd −eyd) . 2 2 Using the fact that min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) } ≥ M ≥ 1 we obtain 2 2 (cid:110) (cid:111) M˜ := max (cid:107)Λ(x)−Λ(y)(cid:107) ,(cid:107)Λ(x)−Λ(y)(cid:107)β 2 2 (cid:40) (cid:41) (cid:107)x−y(cid:107) (cid:107)x−y(cid:107)β ≤ max c π 2 ,cβπβ 2 4 min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) } 4 min{(cid:107)x(cid:107)β,(cid:107)y(cid:107)β} 2 2 2 2 (cid:110) (cid:111) max (cid:107)x−y(cid:107) ,(cid:107)x−y(cid:107)β 2 2 ≤ c π . 4 (cid:110) (cid:111) min (cid:107)x(cid:107)β,(cid:107)y(cid:107)β 2 2 10 PATRICK COMDU¨HR This yields together with (2.6), (2.8), (cid:107)x+(0,...,0,a)(cid:107) ≤ (cid:107)x(cid:107) +a and replacing 2 2 x and y by Λ(x) and Λ(y) (cid:107)DF(Λ(x))−DF(Λ(y))(cid:107) (cid:16) (cid:17) ˜ ˜ ≤ CC (min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) }+a)·M +|(cid:107)x(cid:107) −(cid:107)y(cid:107) | 2 2 2 2   min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) }+a (cid:110) (cid:111) ≤ CC˜c4π (cid:110) 2 2 (cid:111) +1·max (cid:107)x−y(cid:107)2,(cid:107)x−y(cid:107)β2 min (cid:107)x(cid:107)β,(cid:107)y(cid:107)β 2 2   min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) }+a (cid:110) (cid:111) ≤ CC˜c4π (cid:110) 2 2 (cid:111) +1·max (cid:107)x−y(cid:107)2,(cid:107)x−y(cid:107)β2 . min (cid:107)x(cid:107)β,(cid:107)y(cid:107)β 2 2 (cid:110) (cid:111) Since min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) } ≥ min (cid:107)x(cid:107)β,(cid:107)y(cid:107)β ≥ 1 and 2 2 2 2 (cid:110) (cid:111) min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) }+a+min (cid:107)x(cid:107)β,(cid:107)y(cid:107)β min{(cid:107)x(cid:107) ,(cid:107)y(cid:107) }+a 2 2 2 2 2 2 +1 = (cid:110) (cid:111) (cid:110) (cid:111) min (cid:107)x(cid:107)β,(cid:107)y(cid:107)β min (cid:107)x(cid:107)β,(cid:107)y(cid:107)β 2 2 2 2 (cid:110) (cid:111) ≤ (2+a)·min (cid:107)x(cid:107)1−β,(cid:107)y(cid:107)1−β , 2 2 there exists c > 0 such that we have for all x,y ∈ H 7 ≥M (cid:107)DF(Λ(x))−DF(Λ(y))(cid:107) (3.9) (cid:110) (cid:111) (cid:110) (cid:111) ≤ c min (cid:107)x(cid:107)1−β,(cid:107)y(cid:107)1−β ·max (cid:107)x−y(cid:107) ,(cid:107)x−y(cid:107)β , 7 2 2 2 2 which proves the Lemma. (cid:3) Lemma 3.3. For all k ∈ N and t ∈ (0,∞) we have (cid:13) (cid:13) (cid:13) d d (cid:13) (cid:0) (cid:1) (cid:13) (L ◦φ )(t)− (L (0,...,0,Ek(t)+M))(cid:13) ≤ I (t)+I (t) ·(Ek)(cid:48)(t) k−1 k k−1 1,k 2,k (cid:13)dt dt (cid:13) 2 with ac 4 I (t) := 1,k Ek+1(t)·Ek(t) and 2(cid:107)s (cid:107) +1 (3.10) I (t) := c · k 2 , 2,k 8 Ek(t)1+β where c := c2c (d+log(M +a)+M). 8 4 7