Math. Nachr. 280, No. 1–2, 140 – 151 (2007) / DOI 10.1002/mana.200410470
Non-differentiability points of Cantor functions
Wenxia Li∗ Department of Mathematics, East China Normal University, Shanghai 200062, P. R. China
Received 20 July 2004, revised 14 June 2005, accepted 28 July 2005 Published online 19 December 2006
Key words Hausdorff dimension, packing dimension, non-differentiability, Cantor function MSC (2000) Primary: 28A80; Secondary: 28A78 Sr Let the Cantor set C in R be defined by C = j=0 hj (C) with a disjoint union, where the hj ’s are similitude mappings with ratios 0
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
The Cantor set C in R is defined as the unique nonempty compact set invariant under hj’s: r C = hj(C), (1.1) j=0 where hj(x)=ajx + bj, j =0, 1,...,r, with 0 As usual, the elements of C in (1.1) can be encoded by digits in Ω={0, 1,...,r} as follows. We write ω {σ σ ,σ ,... σ j ∈ } ∗ ∞ k k {σ σ ,σ ,...,σ k Ω = =( (1) (2) ): ( ) Ω and Ω = k=1Ω with Ω = =( (1) (2) ( )) : σ(j) ∈ Ω} for k ∈ N. |σ| is used to denote the length of a word σ ∈ Ω∗.Foranyσ, τ ∈ Ω∗ write σ ∗ τ = (σ(1),...,σ(|σ|),τ(1), ...,τ(|τ|)), and write τ ∗ σ =(τ(1),...,τ(|τ|),σ(1),σ(2),...) for any τ ∈ Ω∗, ω ω σ ∈ Ω . σ|k =(σ(1),σ(2),...,σ(k)) for σ ∈ Ω and k ∈ N. Denote hσ(x)=hσ(1) ◦···◦hσ(k)(x) for k k σ ∈ Ω and x ∈ R. Then for σ ∈ Ω , the intervals hσ∗0([0, 1]),hσ∗1([0, 1]),...,hσ∗r([0, 1]) are contained in hσ([0, 1]) in this order where the left endpoint of hσ∗0([0, 1]) coincides with the left endpoint of hσ([0, 1]),and h , h , the right endpoint of σ∗r([0 1]) coincides with the right endpoint of σ([0 1]). Moreover the length of interval h , m h , k a a σ ∈ k m · σ([0 1]) equals ( σ([0 1])) = j=1 σ(j) =: σ for Ω ,where ( ) denotes the one-dimensional Lebesgue measure. For j =1, 2,...,wedefine Cj = hσ([0, 1]). σ∈Ωj ∗ e-mail: [email protected], Phone: +00 86 21 5268 2621, Fax: +00 86 21 5268 2621 c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Math. Nachr. 280, No. 1–2 (2007) 141 C ↓ C j →∞ x ∈ C σ ∈ ω {x} ∞ h , Then j as and can be encoded by a unique Ω satisfying = k=1 σ|k([0 1]). We usually denote this unique code of x by x and use x(k) to denote the k-th component of x . The endpoints of ∗ hσ([0, 1]) for a σ ∈ Ω will be called the endpoints of C. So the set of endpoints of C is countable. Obviously any endpoint e of C lies in C and except for a finite number of terms, its coding e consists of either only the symbol 0 if e is the left endpoint of some hσ([0, 1]), or only the symbol r if e is the right endpoint of some hσ([0, 1]). Let µ be the self-similar Borel probability measure on C corresponding to the probability vector p ,p ,...,p p > r p ( 0 1 r) where each i 0 and i=0 i =1, i.e., the measure satisfying r −1 µ(A)= pjµ hj (A) for any Borel set A, j=0 and so k k µ(hσ([0, 1])) = pσ(j) =: pσ, for any σ ∈ Ω ,k∈ N. (1.3) j=1 Obviously, µ is atomless. Consider the distribution function of the probability measure µ, also called Cantor function or a self-affine “devil’s staircase” function: F (x)=µ([0,x]),x∈ [0, 1]. (1.4) Then F (x) is a non-decreasing continuous function with F (0) h x ax − a j j , ,...,r, (R3) For a general Cantor set C defined by (1.1), let ξ be the Hausdorff dimension of C which is determined ξ by (1.2). Taking pj = aj ,j=0, 1,...,r, Dekking & Li [3] showed that 2 2 dimH S = ξ =(dimH C) . It is important to note that the results for all three cases above present a curious property, i.e., 2 dimH S =(dimH C) . (1.5) In fact, dimH S depends on the choice of µ supported by C. For all three cases above the measures µ are www.mn-journal.com c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 142 Wenxia Li: Non-differentiability points of Cantor functions ξ constructed in the same way by choosing pi = ai ,i=0, 1,...,r, so that the corresponding measures µ are of ξ-Ahlfors regularity,i.e.,thereexistc1,c2 > 0 such that ξ ξ c1δ ≤ µ(B(x, δ)) ≤ c2δ for all x ∈ C and 0 <δ<1,whereB(x, δ) is the interval of centre x and length 2δ. The curious property shown in (1.5) has been revealed successfully by K. J. Falconer in his recent paper [4] for a very general nonempty compact subset E (unnecessarily restricted to Cantor sets in (1.1)) of R and the positive finite Borel d-Ahlfors regular measures µ supported by E. Then a natural question one is interested in is that whether or not the dimension formula shown in (1.5) still holds if the measure µ is not of ξ-Ahlfors regularity. The following (R4) gives a negative answer. (R4)Letr =1.Takep = a0 and p = a1 . Then the corresponding probability measure µ defined 0 a0+a1 1 a0+a1 + − by (1.3) is a non-Ahlfors regular measure except for a0 = a1.Letd and d be respectively determined by a − a a + + a − a a + log 1 log( 0 + 1) ad ad − log 1 log( 0 + 1) ad , log 0 + 1 + 1 log 1 =0 log a1 log a1 and a − a a − − a − a a − log 0 log( 0 + 1) ad ad − log 0 log( 0 + 1) ad . log 0 + 1 + 1 log 0 =0 log a0 log a0 + − Morris [7] proved that dimH S =max{d ,d }. In the present paper we focus on the Cantor sets C in (1.1) and consider a wide variety of self-similar Borel probability measures µ in (1.3) where we only require that pi >ai, i =0, 1,...,r. Thus these measures µ fail to be Ahlfors regular for general choices of pi’s and include all above cases (R1–4) considered in [1, 2, 3, 7]. The requirement that pi >ai, i =0, 1,...,r, ensures that the upper derivative of F (x) is infinite for all x ∈ C,the proof of which is given in the next section as Proposition 2.1. Thus S now can be characterized as S = N + ∪ N − ∪{endpoints of C}, where N + (N −) is the set of non-endpoints of C at which the lower right (left) derivative of F (x) is finite. Thus + − ∗ ∗ dimH S =max{dimH N , dimH N }. For each δ>0 we construct sets C (q1(δ)),C(q2(δ)),C (q3(δ)) and + C(q4(δ)), to be defined later by (2.10), (2.11), (2.16), (2.17), (2.19) and (2.20) respectively, to approximate N and N − by ∗ + ∗ − C (q1(δ)) ⊆ N ⊆ C(q2(δ)),C(q3(δ)) ⊆ N ⊆ C(q4(δ)). Therefore, ∗ + η(q1(δ)) := dimH C (q1(δ)) ≤ dimH N ≤ η(q2(δ)) := dimH C(q2(δ)), and ∗ − η(q3(δ)) := dimH C (q3(δ)) ≤ dimH N ≤ η(q4(δ)) := dimH C(q4(δ)), where η(q1(δ)) and η(q2(δ)) are determined by (2.23) and (2.24) respectively, and η(q3(δ)) and η(q4(δ)) + are determined in the same way just to replace aδ by aδ there. We prove that dimH N = limδ→+∞ η(q1(δ)) − and dimH N = limδ→+∞ η(q3(δ)) by showing that limδ→+∞ η(q1(δ)) = limδ→+∞ η(q2(δ)) and limδ→+∞ η(q3(δ)) = limδ→+∞ η(q4(δ)). By means of monotonicity of η(·) and (2.18) we obtain + − Theorem 1.1 Suppose that pi >ai, i =0, 1,...,r.Letd (δ) and d (δ) be respectively given by p + p + log r d (δ) log r d (δ) log aσ + 1 − log aδ =0, log ar log ar σ∈Ωδ and p − p − log r d (δ) log r d (δ) log aσ + 1 − log aδ =0, log ar log ar σ∈Ωδ c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com Math. Nachr. 280, No. 1–2 (2007) 143 where for each δ>0, Ωδ, aδ and aδ are defined by (2.13), (2.14) and (2.15) respectively. Then we have [A] dimP S =dimB S = ξ,whereξ is defined in (1.2); + − + + − [B] dimH S =max{dimH N , dimH N },wheredimH N = limδ→+∞ d (δ) and dimH N = − limδ→ ∞ d (δ). + r −1 pi pi ai ai i , ,...,r ,i A very interesting and simple case occurs if = ( i=0 ) , =01 . In this case, all ai = , ,...,r, δ> k k k δ a ak a ak 0 1 are equal. Thus for each 0, Ωδ =Ω for some = ( ) by (2.13), and so δ = r and δ = 0 . Theorem 1.1 then leads to (or alternatively from Corollary 2.3, Lemma 2.4 and Lemma 2.5 directly) p a r a −1 i , ,...,r d+ d− Corollary 1.2 Let i = i ( i=0 i) , =0 1 .Let and be respectively determined by r r a − a + a − a + log r log i=0 i d log r log i=0 i d log aj + 1 − log ar =0 log ar log ar j∈Ω and r r log a − log ai − log a − log ai − 0 i=0 ad − 0 i=0 ad . log j + 1 log 0 =0 log a0 log a0 j∈Ω Then [A] dimP S =dimB S = ξ,whereξ is defined in (1.2); + − + + − − [B] dimH S =max{dimH N , dimH N },wheredimH N = d and dimH N = d . This also was given by Morris [7] for r =1(see R4). ξ We like to remark that the case where pi = ai , i =0, 1,...,r, is not so simple as the case in Corollary 1.2 except all ai = a which reduces to the case considered in [2] and satisfies the conditions in Corollary 1.2. A finer + − + − 2 treatment in [3] shows that dimH N =dimH N = limδ→+∞ d (δ) = limδ→+∞ d (δ)=ξ (see R3). The proof of Theorem 1.1 is given in the next section. 2 Proofs In this section, we explore the Hausdorff, packing and box dimensions of S for the case where pi >ai, i = 0, 1,...,r. Proposition 2.1 Let µ and F (x) be given by (1.3) and (1.4) respectively. Let pi >ai,i=0, 1,...,r.Then the upper right derivative of F (x) at a non-right-endpoint t of C is infinite and the upper left derivative of F (x) at a non-left-endpoint t of C is infinite. Thus the upper derivative of F (x) is infinite for all x ∈ C. Proof. Forany t in C, t not a right endpoint, let its code be t =(t(1),t(2),...).Then t has infinitely many entries lying in Ω \{r}. Suppose t has an entry from Ω \{r} in position j.Thent lies in the interval h , u h , et|(j−1)([0 1]) but is not equal to the right endpoint of et|(j−1)([0 1]),where u =(t(1),...,t(j − 1),r,r,...). Note that u is also the right endpoint of hue|j([0, 1]) and that t/∈ hue|j([0, 1]). Thus we have that t, u ∈ h , t, u ⊇ h , P t, F t et|(j−1)([0 1]) and ( ] ue|j([0 1]). Consider the slope of the line segment from the point =( ( )) on the graph of F (x) to the point Q =(u, F (u)).Wehave F (u) − F (t) µ((t, u]) µ hue|j([0, 1]) pet| j− pr = ≥ = ( 1) . (2.1) u − t u − t m h , a et|(j−1)([0 1]) et|(j−1) Since pi >ai,i =0, 1,...,r, it is directly derived from (2.1) that the upper right derivative of F (x) at a non-right-endpoint t of C is infinite. Symmetrically the upper left derivative of F (x) at a non-left-endpoint t of C is infinite. From Proposition 2.1 it follows that for the case where pi >ai, i =0, 1,...,r,thesetS can be decomposed into S = N + ∪ N − ∪{endpoints of C}, (2.2) www.mn-journal.com c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 144 Wenxia Li: Non-differentiability points of Cantor functions where N + (N −) is the set of non-endpoints of C at which the lower right (left) derivative of F (x) is finite. The following lemma characterizes the elements of N + and N − by means of property of their codes. Lemma 2.2 Suppose that pi >ai, i =0, 1,...,r.LetΓ={0, 1,...,r− 1}.Lett ∈ C be no endpoint of pi C z t, n n t w i∈ and let ( ) denote the position of the -th occurrence of elements of Γ in .Let =min Γ ai and let pi w =maxi∈ .Then Γ ai + z(t,n+1) log ar pr n (I) if t ∈ N ,thenlim supn→∞ z t,n − p + log w − log a z t,n p ≥ 0; ( ) log r r ( )log r z t,n a p (II) if t ∈ C satisfies lim sup ( +1) − log r + log w − log r n > 0,thent ∈ N +. n→∞ z(t,n) log pr ar z(t,n)logpr { , ,...,r} Symmetrically if we replace Γ by 1 2 ,then − z(t,n+1) log a0 p0 n (I’) if t ∈ N ,thenlim supn→∞ z t,n − p + log w − log a z t,n p ≥ 0; ( ) log 0 0 ( )log 0 z t,n a p if t ∈ C satisfies lim sup ( +1) − log 0 + log w − log 0 n > 0,thent ∈ N −. (II’) n→∞ z(t,n) log p0 a0 z(t,n)logp0 P r o o f. Since the upper derivative of F (x) is infinite for all x ∈ C by Proposition 2.1, N + (N −) consists of non-endpoints of C at which the lower right (left) derivative of F (x) is finite. For the demonstration of the statement (I), it suffices to show that the lower-right derivative of F (x) is infinite at a non-endpoint t of C when z(t, n +1) log ar pr n lim sup − + log w − log < 0. (2.3) n→∞ z(t, n) log pr ar z(t, n)logpr Consider such a point t with t =(t(1),t(2),...). By (2.3) let k be a positive integer such that z(t, n +1) log ar pr n − + log w − log F x − F t pe pr ( ) ( ) ≥ t|j x − t a et|(i−1) z(t,n+1)−1 pr i pt i = =1 ( ) z(t,n)−1 a i=1 t(i) z (t,n) p z(t,n+1)−z(t,n) t(i) (2.5) = at(z(t,n))pr at i i=1 ( ) ⎛ ⎞ ⎛ ⎞ 1 z(t,n) z(t,n) z(t,n) z(t,n+1) − p ⎜ z(t,n) 1⎝ t(i) ⎠ ⎟ ≥ min ai ⎝pr ⎠ . 0≤i≤r at i i=1 ( ) Let c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com Math. Nachr. 280, No. 1–2 (2007) 145 ⎛ ⎞ 1 z(t,n) z(t,n) z(t,n+1) − p z(t,n) 1⎝ t(i) ⎠ Q = pr at i i=1 ( ) ⎛ ⎞ 1 z(t,n)−n z(t,n) z(t,n) z(t,n+1) − p z(t,n) p z(t,n) 1 r ⎝ t(i) ⎠ = pr ar at(i) i=1,t(i)= r z(t,n)−n z(t,n+1) − z(t,n) n z(t,n) 1 pr ≥ pr w z(t,n) . ar Taking logs and by (2.4) we have z(t, n +1) log ar pr n log Q ≥ − + log w − log log pr >qlog pr > 0. (2.6) z(t, n) log pr ar z(t, n)logpr Since t is a non-endpoint, z(t, n) →∞as n →∞and the lower-right derivative of F (x) is infinite at t by (2.5) and (2.6). For the proof of statement (II) let t ∈ C be such that z(t, n +1) log ar pr n lim sup − + log w − log > 0. n→∞ z(t, n) log pr ar z(t, n)logpr Then there exists a sequence {nk} of positive integers such that z(t, nk +1) log ar pr nk − + log w − log >c (2.7) z(t, nk) log pr ar z(t, nk)logpr c xk h e , jk z t, nk − for some positive constant .Let be the left endpoint of (t|jk)∗(t(jk +1)+1)([0 1]),where = ( ) 1. x k t ,...,t jk ,t jk , ,..., ,... uk he , Thus we have =((1) ( ) ( +1)+1 0 0 ).Let be the right endpoint of t|(jk+1)([0 1]). u k t ,...,t jk ,t jk ,r,r,r,... uk,xk he , Then =((1) ( ) ( +1) ). Thus ( ) is the gap on the right side of t|(jk +1)([0 1]) m uk,xk xk − uk a β βj j , ,...,r − and (( )) = = et|jk t(jk +1) where by , =01 1, we denote the length of the h , h , t, x ⊇ u ,x µ t, x µ t, u gap between the images j([0 1]) and j+1([0 1]). Note that [ k] [ k k] and (( k]) = (( k]) + µ uk,xk µ t, uk ≤ µ he , t z t, nk − u k| z t, nk − (( ]) = (( ]) t|(z(t,nk+1)−1)([0 1]) since ( ( +1) 1) = ( ( +1) 1). Therefore we have F xk − F t µ t, xk ≤ µ he , pe , ( ) ( )= (( ]) t|(z(t,nk+1)−1)([0 1]) = t|(z(t,nk+1)−1) and xk − t ≥ m uk,xk ae β . ([ ]) = t|(z(t,nk)−1) t(z(t,nk)) ∗ Denote β∗ =minj∈{0,1,...,r−1} βj and a =maxj∈{0,1,...,r} aj. Then we obtain with a similar reasoning which led to (2.5) F x − F t pe ( k) ( ) ≤ t|(z(t,nk+1)−1) xk − t ae β t|(z(t,nk)−1) t(z(t,nk)) z( t,nk) at z t,n pt i ( ( k)) z(t,nk+1)−z(t,nk) ( ) = pr βt z t,n pr at i ( ( k)) i=1 ( ) (2.8) ⎛ ⎞ ⎛ ⎞ 1 z(t,nk) z(t,n ) ∗ z(t,n +1) z(t,nk) k a k −1 p ⎜ z(t,nk) t(i) ⎟ ≤ ⎝pr ⎝ ⎠ ⎠ . β∗pr at i i=1 ( ) www.mn-journal.com c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 146 Wenxia Li: Non-differentiability points of Cantor functions Let ⎛ ⎞ 1 z t,n z(t,nk) z(t,nk+1) ( k) −1 pt i z(t,nk) ⎝ ( ) ⎠ Q = pr at i i=1 ( ) ⎛ ⎞ 1 z(t,nk)−nk z t,n z(t,nk) z(t,nk+1) ( k ) −1 pr z(t,nk) pt i z(t,nk) ⎝ ( ) ⎠ = pr ar at(i) i=1,t(i)= r z(t,nk)−nk z(t,n +1) k −1 p z(t,nk) nk z(t,nk) r ≤ pr w z(t,nk) . ar Taking logs and using (2.7) we obtain z(t, nk +1) log ar pr nk log Q ≤ − + log w − log log pr Then we have dimH C(q)=η and dimP C(q)=dimB C(q)=dimH C = ξ where ξ is defined in (1.2). It is easy to verify that η(q) is strictly increasing and continuous in q and that η(0+) <η(q) ≤ η(1) = ξ.We also consider for 0 c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com Math. Nachr. 280, No. 1–2 (2007) 147 pi pi >ai,i , ,...,r { , ,...,r− } w i∈ Lemma 2.5 Suppose that =01 .LetΓ= 0 1 1 .Let =min Γ ai and let pi w i∈ =max Γ ai .Let a a p q−1 log r , log r − w − r 1 , 1 =max log log log pr log pr ar log pr and a a p q−1 log r , log r − w − r 1 . 2 =min log log log pr log pr ar log pr ∗ + + Then C (q1) ⊆ N ⊆ C(q2) and η(q1) ≤ dimH N ≤ η(q2). ∗ − − Symmetrically if we replace Γ by {1, 2,...,r},thenC (q3) ⊆ N ⊆ C(q4) and η(q3) ≤ dimH N ≤ η(q4) where a a p q−1 log 0 , log 0 − w − 0 1 , 3 =max log log log p0 log p0 a0 log p0 and a a p q−1 log 0 , log 0 − w − 0 1 . 4 =min log log log p0 log p0 a0 log p0 P r o o f. We only prove the result on N + since the other result on N − can be obtained in the same way. Thus ∗ + + it suffices to prove C (q1) ⊆ N ⊆ C(q2).Lett ∈ N . By Lemma 2.2 (I) we have z(t, n +1) log ar pr n lim sup − + log w − log ≥ 0. n→∞ z(t, n) log pr ar z(t, n)logpr pr z(t,n+1) log ar n z(t,n+1) Now if w ≥ ,thenlim supn→∞ ≥ . If not, we use that ≤ 1 and so lim supn→∞ ≥ ar z(t,n) log pr z(t,n) z(t,n) a p log r − log w − log r 1 .Sowefindthat log pr ar log pr z t, n ( +1) ≥ q−1, lim sup 2 n→∞ z(t, n) ∗ + i.e., t ∈ C(q2). On the other hand, t ∈ C (q1) implies in a similar way that t ∈ N . − r 1 pr p0 pi ai ai i , ,...,r, w w Note that for the special case where = i=0 , =01 we have = = ar = a0 = − r 1 log pr log p0 ai so that q = q = and q = q = . Thus Lemma 2.5 gives the exact Hausdorff i=0 1 2 log ar 3 4 log a0 + − dimensions of N and N . But for general cases we have q1 = q2 and q3 = q4. So Lemma 2.5 fails to + − determine the exact Hausdorff dimensions of N and N . On the other hand, note that both |q1 − q2| and pi pi |q − q | i∈ − i∈ / {| p |, pr|} 3 4 will be very small if log max Ω ai log min Ω ai min log 0 log is small enough, so that η(q1) and η(q3) will be close to η(q2) and η(q4), respectively, since η(q) is continuous in q. To realize this idea we review C as the invariant set under a class of similitude mappings each of which is a composition of certain hj’s, so that Lemma 2.5 could give η(q1) and η(q2)(η(q3) and η(q4)) as close as expected. pi 2 wi i ∈ w i∈ wi δ>w Let = ai , Ω,andlet =max Ω .Let ,andlet ∗ Ωδ = {σ ∈ Ω : wσ ≥ δ and wσ|(|σ|−1) <δ}, (2.13) w |σ| w σ ∈ ∗ σ, σ ∈ σ j r j , ,...,|σ| where σ = j=1 σ(j) for Ω . There exist unique Ωδ with ( )= for =12 and σ (j)=0for j =1, 2,...,|σ | respectively. We denote these two special elements by σδ and σ δ respectively. σ r h , , Note that δ plays the same role in Ωδ as in Ω, in the sense that σδ ([0 1]) is the most right interval in [0 1] hσ , σ δ δ hσe , of the intervals ( ([0 1]))σ∈Ωδ , while plays the same role in Ω as 0 in Ω, in the sense that δ ([0 1]) is the , hσ , most left interval in [0 1] of the intervals ( ([0 1]))σ∈Ωδ .Let |σδ| pr \{σ } p p p|σδ | w w , Γδ := Ωδ δ ; δ := σδ = r ; δ := σδ = |σδ| ar and www.mn-journal.com c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 148 Wenxia Li: Non-differentiability points of Cantor functions k pr a a a|σδ | |σ | k ∈ N ≥ δ . δ := σδ = r where δ =min : (2.14) ar Symmetrically let p|σeδ| \{σ } p p p|σeδ | w w 0 Γδ := Ωδ δ ; δ := σeδ = ; δ := σeδ = 0 a|σeδ| 0 and p k a a a|σeδ | |σ | k ∈ N 0 ≥ δ . δ := σeδ = 0 where δ =min : (2.15) a0 w w w w We denote δ =minσ∈Γδ σ, δ =maxσ∈Γδ σ,andset −1 log aδ log aδ pδ 1 q1(δ)= max , − log wδ − log log pδ log pδ aδ log pδ −1 (2.16) log ar log ar pδ 1 = max , − log wδ − log , log pr log pr aδ log pδ and −1 log aδ log aδ pδ 1 q2(δ)= min , − log wδ − log log pδ log pδ aδ log pδ −1 (2.17) log ar log ar pδ 1 = min , − log wδ − log , log pr log pr aδ log pδ a a since log δ = log r . It is easy to check that log pδ log pr log pr q1(δ) ≤ ≤ q2(δ). (2.18) log ar w =min e wσ wδ =max e wσ Symmetrically we denote δ σ∈Γδ , σ∈Γδ ,andset −1 log aδ log aδ p δ 1 q3(δ)= max , − log wδ − log log p δ log p δ aδ log p δ (2.19) a a p −1 log 0 log 0 δ 1 = max , − log wδ − log . log p0 log p0 aδ log p δ and −1 log aδ log aδ p δ 1 q4(δ)= min , − log w δ − log log p δ log p δ aδ log p δ −1 (2.20) log a0 log a0 p δ 1 = min , − log w δ − log , log p0 log p0 aδ log p δ ea a p log δ = log 0 q (δ) ≤ log r ≤ q (δ). since log peδ log p0 .Alsowehave 3 log ar 4 Now we are ready to prove Theorem 1.1. + + P r o o f o f T h e o r e m 1.1. By Lemmas 2.4 and 2.5, [A] is trivial since we have dimB N =dimP N = − − + − dimB N =dimP N =dimH C = ξ.SinceS = N N {the endpoints of C} in our case, we have + − + dimH S =max{dimH N , dimH N }. In the following we only determine the Hausdorff dimension of N because of the symmetry between N + and N −. Note that for each σ ∈ Ωδ we have δ ≤ wσ c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com Math. Nachr. 280, No. 1–2 (2007) 149 | log wσ − log wτ |≤log w. (2.21) Now let Hδ = {hσ : σ ∈ Ωδ}. Note that hσ is a similitude mapping with ratio 0 σ∈Hδ ξ ξ a µδ We also have that defined in (1.2) satisfies σ∈Ωδ σ =1. If we denote the self-similar probability measure on C corresponding to the probability vector (pσ : σ ∈ Ωδ),thenµδ = µ since µδ(hτ ([0, 1])) = µ(hτ ([0, 1])) k + − for any τ ∈ Ωδ and k ∈ N. Hence for the corresponding sets Nδ and Nδ of non-differentiability points we have + + − − Nδ = N and Nδ = N for all δ>0.ThenbyLemmas2.4and2.5withΩ replaced by Ωδ, Γ by Γδ, ar by aδ, pr by pδ, w by wδ, w by wδ,andq1,q2 by q1(δ), q2(δ) respectively, we have + + η(q1(δ)) ≤ dimH Nδ =dimH N ≤ η(q2(δ)), (2.22) where by means of (2.11), (2.12) and Lemma 2.4 η(q1(δ)) and η(q2(δ)) are given by η(q1(δ)) η(q1(δ)) q1(δ)log aσ +(1− q1(δ)) log aδ =0 (2.23) σ∈Ωδ and η(q2(δ)) η(q2(δ)) q2(δ)log aσ +(1− q2(δ)) log aδ =0. (2.24) σ∈Ωδ + From the analysis above it is expected that limδ→+∞ η(q1(δ)) = limδ→+∞ η(q2(δ)) = dimH Nδ since pδ 1 pδ 1 lim log wδ − log = lim log wδ − log =0, δ→+∞ aδ log pδ δ→+∞ aδ log pδ by (2.21), leading to log pr lim q1(δ) = lim q2(δ)= ∈ (0, 1). (2.25) δ→+∞ δ→+∞ log ar Now we rewrite (2.23) and (2.24) as η(q1(δ)) q δ aσ 1( ) log σ∈Ωδ η(q1(δ)) = · (2.26) q1(δ) − 1 |σδ| log ar and η(q2(δ)) q δ aσ 2( ) log σ∈Ωδ η(q2(δ)) = · . (2.27) q2(δ) − 1 |σδ| log ar Suppose that limδ→+∞(η(q2(δ)) − η(q1(δ))) = 0 doesn’t hold. Then there exist a positive real number 0 and a sequence of positive real numbers δi ↑ +∞ such that η(q2(δi)) − η(q1(δi)) >0. (2.28) www.mn-journal.com c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 150 Wenxia Li: Non-differentiability points of Cantor functions Note that η(q2(δi)) η(q1(δi)) log σ∈ aσ log σ∈ aσ Ωδi − Ωδi |σδ | log ar |σδ | log ar i i η(q2(δi)) aσ 1 σ∈Ωδ = log i |σ | a η(q1(δi)) δi log r aσ σ∈Ωδ i η(q1(δi)) η(q2(δi))−η(q1(δi)) σ∈ aσ aσ 1 Ωδi (2.29) = log η q δ |σδ | ar ( 1( i)) i log σ∈ aσ Ωδi log(maxσ∈ aσ) 0 ≥ Ωδi |σδi | log ar a 0 log maxσ∈Ωδ σ = i . log aδi Hence by (2.26), (2.27), (2.28) and (2.29) we have 0 <η(q2(δi)) − η(q1(δi)) η(q2(δi)) η(q1(δi)) aσ aσ q (δi) log σ∈Ωδ q (δi) log σ∈Ωδ 2 · i − 1 · i = q δ − |σ | a q δ − |σ | a 2( i) 1 δi log r 1( i) 1 δi log r ⎛ ⎞ η(q2(δi)) η(q1(δi)) aσ aσ q (δi) log σ∈Ωδ log σ∈Ωδ 2 ⎝ i − i ⎠ = q δ − |σ | a |σ | a 2( i) 1 δi log r δi log r η(q1(δi)) aσ log σ∈Ωδ q (δi) q (δi) i 2 − 1 + |σ | a q δ − q δ − δi log r 2( i) 1 1( i) 1 η(q1(δi)) a aσ q (δi) 0 log maxσ∈Ωδ σ log σ∈Ωδ q (δi) q (δi) ≤ 2 · i + i 2 − 1 q (δi) − 1 log aδ |σδ | log ar q (δi) − 1 q (δi) − 1 2 i i 2 1 a q (δi) 0 log maxσ∈Ωδ σ η(q (δi))(q (δi) − 1) q (δi) q (δi) 2 · i 1 1 2 − 1 . = q δ − a + q δ q δ − q δ − 2( i) 1 log δi 1( i) 2( i) 1 1( i) 1 Note that in the last line above the first term is negative and the second term tends to zero when i is large enough by (2.25), yielding a contradiction! Therefore we have limδ→+∞(η(q2(δ)) − η(q1(δ))) = 0. Thus + + limδ→+∞(η(q2(δ)) = limδ→+∞ η(q1(δ))) = dimH Nδ =dimH N by (2.22). Symmetrically we have η q δ N − N − η · η q δ ≤ lim δ→+∞ ( ( 3( )) = dimH δ =dimH . Finally, by means of monotonicity of ( ) we have ( 1( )) log pr + + + − η = d (δ) ≤ η(q (δ)) by (2.18). This gives dimH N = limδ→ ∞ d (δ),anddimH N = log ar 2 + − limδ→+∞ d (δ) in the same way. Acknowledgements The author would like to thank the anonymous referees for their valuable comments and suggestions that lead to the improvement of the manuscript. This project is supported by the National Science Foundation of China #10371043 and Shanghai Priority Academic Discipline. References [1] R. Darst, The Hausdorff dimension of the non-differentiability set of the Cantor function is [ln(2)/ln(3)]2, Proc. Amer. Math. Soc. 119, 105–108 (1993). [2] R. Darst, Hausdorff dimension of sets of non-differentiability points of Cantor functions, Math. Proc. Camb. Phi. 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