Epsilon-Distortion Complexity for Cantor Sets

ε -Disto r t ion Complexit y for Can tor Sets C. Bonanno Dip artimento di Matematic a Applic ata Universit` a di Pisa 56127 Pisa, Italy J.-R. Chazottes, P . Collet Centr e de Physique Th´ eorique Ec ole p o l yte chnique, CNRS 91128 Palaise au, F r anc e No ve mber 21, 2018 Abstract W e define the ε -distor tion complexit y of a set as the shortest pro - gram, running on a universal T uring machine, which pro duces this set at the precision ε in the s ense of Hausdorff distance. Then, w e esti- mate the ε -dis tortion co mplexity of v arious central Ca n to r sets on the line g enerated by itera ted function systems (IFS’s). In particular, the ε -distortion complexity o f a C k Cantor set dep ends, in general, o n k and on its box c o unt ing dimension, contrarily to Ca nt o r sets genera ted by p olynomia l IFS or random a ffine Cantor s e ts. k eywords : central Cantor sets, iterated function system, random Can- tor sets, scaling function, C k Cantor sets, b ox co un ting dimension. 1 In tro d u ction No wa d a ys, compu ters are b eing wid ely used to generate images in the anal- ysis and sim u lations of real-life p ro cesses and their mathematical mo d els. A natural issue is to measure the complexit y of dra w ing a set of p oin ts on a computer, whic h d escrib es a con tinuous ob ject at a giv en precision. Partic u- lar examples of complex ob jects are fractal sets whic h arise in many con texts [3]. W ell-kno wn examples of f r actal sets are strange attractors of d issipativ e dynamical systems and Julia sets. Another wa y to generate fr actal s ets is to use iterated fu nction systems [2]. The w ay w e measure the complexit y of a (compact) set can b e collo- quially describ ed as follo ws. W e define the ε -distortion complexit y of a set C as the minimal length of the programs pro ducing a finite set ε -close to 1 C , in the s ense of Hausdorff distance. As in the classical notion of Kol- mogoro v complexit y of sequences, by programs we mean programs r unnin g on a univ ersal T ur ing m ac hine [11]. W e are interested in the b eh avior of the ε -distortion complexit y wh en ε is getting small, and in ev entual relations of this b eha vior with other characte r istics of th e set ( e.g. , fr actal dimension). In the present article w e consider v arious classes of C an tor sets on the real line generated by iterated fun ction systems (IFS’s) [2] and compute b ound s f rom ab o ve and b elo w of their ε -distortion complexit y as a function of ε . W e first consider IFS’s with p olynomial con tr actions and obtain the upp er b ound const × log( ε − 1 ) for the ε -distortion complexit y of the generated Can tor s et, where th e (finite) constan t ma y dep end on the p olynomials. W e can pro d uce “man y” p olynomial IFS’s with a lo wer b ound of the same order using a probabilistic construction. It turns out that some particular C an tor sets lik e th e usual midd le third Cantor set are of muc h lo wer complexit y . F or analytic IFS’s, we obtain th e upp er b ound const × (log( ε − 1 )) 2 . Next w e consid er random cen tral C an tor sets pro duced by affine IFS’s, for w hic h the con traction rate is c hosen at random at eac h step of the constru ction. In this case, we get the u pp er b ound const × (log( ε − 1 )) 2 and the lo w er b ound const × (log( ε − 1 )) 2 − δ , for an y δ > 0, for almost all such Canto r sets (wher e the constan t in our b oun d d ep ends on δ an d tends to 0 when δ → 0). Finally , w e consider C k IFS’s. Con trarily to the p revious cases, the leading, asymptotic b eha vior of the ε -distortion complexit y dep ends on the b o x count in g dimension D of th e generated Can tor set. Indeed, we obtain the up p er b ound cons t × ε − D k − δ , for an y δ > 0 (where the constant in our b ound dep ends on δ and b lo ws u p wh en δ → 0). W e then constru ct “many” C k (random) Cantor sets with a lo w er b ound const × ε − D k + δ (for any δ > 0), b y constructing their scaling f unction [14]. The case of sets r ed uced to one p oin t on the line was inv estigat ed in [7] where in particular the Hausdorff dimension of the set of reals with give n asymptotic complexity is computed. F or graph s of functions, fr om the p oint of view of determining the v alues of a fun ction at giv en precision, relations with ε -ent rop y are obtained in [1]. Another notion of complexit y is to ask ab out the smallest execution time of the programs generating a giv en set with ε -precision, in the sense of Hausdorff distance [15]. This wa s used in [6] to sho w that a class of Julia sets was p olynomial time compu table. This article is organized as f ollo ws. In S ection 2 we d efine the ε -distortion complexit y of a compact set and state our results. Section 3 is devo ted to the pro ofs. 2 Definitions and r esults F or a compact set C ⊂ R d , we define its ε -distortion c omplexity as follo ws. Definition 2.1. Th e ε - distortion c omplexity of a c omp act se t C ⊂ R d at 2 pr e cision ε > 0 is define d by ∆( C , ε ) = min { ℓ ( P ) : d H ( C ( P ) , C ) < ε } , wher e the minimum is taken over al l binary pr o gr ams P ∈ { 0 , 1 } ∗ running on a universal T uring machine U , which pr o duc e a finite subset C ( P ) ∈ R d ; ℓ ( P ) is the pr o gr am length; d H denotes the Hausdorff distanc e. Notice th at, b ecause of the compactness of C , we can use a minimum in the ab ov e d efinition, which alw a ys leads to a finite num b er. F or the reader’s con v enience, w e recall that the Hausdorff d istance d H b et ween t wo closed subsets F 1 , F 2 of a metric space with metric d is giv en b y (see, e. g. , [12, 2]) d H ( F 1 , F 2 ) = max { sup x 2 ∈ F 2 d ( x 2 , F 1 ) , sup x 1 ∈ F 1 d ( x 1 , F 2 ) } . Remark 2.1. If T is a b i-Lipschitz map, ther e e xi st two p ositive c onstants c 1 , c 2 such that, if C i s a c omp act set, we have ∆( T ( C ) , c 1 ε ) ≤ ∆( C , ε ) ≤ ∆( T ( C ) , c 2 ε ) . W e no w r ecall the definition of Cantor sets generated b y iterated function systems [2]. F or the sake of simplicit y , w e restrict ourselves to Can tor sets in the unit interv al [0 , 1] ⊂ R , although several results can b e easily generalised to arbitrary finite dimension. Let A = [0 , 1] and let I b e a fi nite set of indices with at least t wo elemen ts. An Iter ate d F unction System (IFS for sh ort) is a collectio n { φ i : A → A : i ∈ I } of inj ectiv e con tractions on A with un if orm con traction rate ρ ∈ (0 , 1), an d suc h that φ i ( A ) ∩ φ j ( A ) = ∅ for i 6 = j . F or any infinite w ord ω ∈ I ∞ and for an y n ∈ N , let ω n 1 ∈ I n denote the prefix of length n given by the fi r st n symb ols of ω , and let φ ω n 1 := φ ω n ◦ φ ω n − 1 ◦ · · · ◦ φ ω 1 . (1) The map π : I ∞ → A d efined b y ω 7→ π ( ω ) := T ∞ n =0 φ ω n 1 ( A ) is con tinuous (in pro d uct top ology) and, since diam ( φ ω n 1 ( A )) ≤ ρ n diam ( A ) , π ( ω ) is a p oint in A for all ω ∈ I ∞ . Th e set C := π ( I ∞ ) = [ ω ∈I ∞ ∞ \ n =0 φ ω n 1 ( A ) 3 is a Can tor set and satisfies C = [ i ∈I φ i ( C ) . (2) W e are intereste d in the b ehavio u r of ∆( C , ε ) wh en ε tends to zero. Note that this is a m onotone d ecreasing fun ction. Notation. In the se quel we write f ≍ g if ther e ar e two p ositive c onstants C 1 and C 2 such that f or any ε > 0 smal l enough C 1 f ( ε ) ≤ g ( ε ) ≤ C 2 f ( ε ) . We write f 4 g if ther e is a p ositive c onstant C such that for any ε > 0 smal l enough f ( ε ) ≤ C g ( ε ) . Our fir st result d eals with p olynomial IFS’s. Theorem 2.1. L et C b e a Cantor set gener ate d by an IFS with p olynomial functions. Then ∆( C , ε ) 4 log( ε − 1 ) . (3) Mor e over, for any δ > 0 , ther e exist (many) p olyno mial IFS’s such that the gener ate d Cantor set satisfies (1 − δ ) log ( ε − 1 ) ≤ ∆ ( C , ε ) . (4) Remark 2.2. A mor e pr e ci se upp er b ound fol lows e asily fr om the pr o of, namely ∆( C , ε ) ≤ X i ∈I (1 + deg φ i ) ! log( ε − 1 ) + o (log ( ε − 1 )) . A mor e pr e cise lower b ound of the same kind c an also b e obtaine d f or a lar ge class of Cantor sets which ar e gener ate d by a set of ful l me asur e of some r andom p olynomia l IFS’s (se e b elow for the definition of a r andom IFS). Remark 2.3. The classic al examples of Cantor sets ar e the midd le 1 ρ -th Cantor sets in the unit interval ( ρ = 1 3 gives the usual midd le thir d Cantor set). They c an b e thought of as ge ne r ate d by IFS with affine c ontr actions φ 0 and φ 1 , and c ontr action r ate ρ . F or these Cantor sets, al l we ne e d for their c onstruction is the know le dge of ε and of ρ . We c an cho ose ε to b e a numb er of low c omplexity. In the p articular c ase wher e ρ is r ationa l, ∆( C , ε ) gr ows slower than any unb ounde d p artial r e cursive function. 1 1 W e thank A rnaldo Mandel for this observa t ion. 4 Remark 2.4. Notic e that ther e ar e e xamples of Cantor sets with low ε - distortion c omplexity which c ontain numb ers of high c omplexity. This hap- p ens for example in the midd le thir d Cantor set. Our next result is ab out r eal analytic IFS’s. Theorem 2.2. L et C b e a Cantor set ge ne r ate d b y an IFS with r e al analytic functions. Then ∆( C , ε ) 4  log( ε − 1 )  2 . W e no w tur n to r an d om affin e IFS’s, for whic h at eac h step of the con- struction we consider a random c hoice for the con traction rate. F or the definition we follo w [4]. Let us consid er a family ( λ k ) k ∈ N of in dep endent iden tically d istributed random v ariables w ith v alues in the in terv al (0 , 1). T o eac h sequence λ = ( λ k ) k ∈ N w e asso ciate a Cant or set C in the follo wing w ay . L et C 0 λ := [0 , 1]. W e define J 1 1 ( λ ) :=  0 , λ 1 2  , J 1 2 ( λ ) :=  1 − λ 1 2 , 1  , and C 1 λ := J 1 1 ( λ ) ∪ J 1 2 ( λ ) . In w ords, C 1 λ is obtained by remo ving the central interv al of length (1 − λ 1 ) from C 0 λ . A t the ( k + 1)-st step, we delete from eac h in terv al J k i ( λ ), i = 1 , . . . , 2 k , the cen tr al in terv al of length (1 − λ k +1 ), obtaining 2 k +1 in terv als J k +1 i ( λ ), i = 1 , . . . , 2 k +1 , su c h that | J k +1 i ( λ ) | = 1 2 k +1 k +1 Y h =1 λ h ∀ i = 1 , . . . , 2 k +1 . (5) Then we define C k +1 λ := 2 k +1 [ i =1 J k +1 i ( λ ) . W e call r andom c entr al Cantor set the s et C λ := ∞ \ k =0 C k λ . W e remark that b y construction the b oundary p oin ts ∂ J k i ( λ ) of all the in - terv als J k i ( λ ) are con tained in C λ . The next theorem states that random central Can tor sets n eed more information th an those generated by p olynomial IFS’s. Theorem 2.3. L et C λ b e a r andom c entr al Cantor set as describ e d ab ove. Then, for any λ ∈ (0 , 1) N , ∆( C λ , ε ) 4  log( ε − 1 )  2 . (6) 5 Mor e over, let us assume that the c ommon distribution of the i.i.d. r andom variables ( λ k ) is absolutely c ontinuous, with a density f ( x ) b ounde d ab ove and b elow away fr om zer o. Then, for any δ > 0 , we have  log( ε − 1 )  2 − δ 4 ∆( C λ , ε ) (7) for almost every λ ∈ (0 , 1) N . W e n o w consider Canto r sets w ith a d ifferen tiable structur e. F ollo wing [14], [13] and [5 ], this corr esp onds to Canto r sets generated by C k IFS’s and w e call them C k Can tor s ets. W e sh all recall their con tru ction in S ubsection 3.4. Theorem 2.4. L et k ≥ 1 . F or any δ > 0 , for any C k Cantor set C with b ox c ounting dimension D , we have ∆( C , ε ) 4 ε − D k − δ . (8) Mor e over, for any δ > 0 , ther e exist (many) C k c entr al Cantor se ts C with b ox c ounting dimension, at most D + δ , such that ε − D k + δ 4 ∆( C , ε ) . (9) W e emphasise that in this case the asymptotic b eha viour of the ε - distortion complexit y , when ε tends to zero, d ep ends in general on the reg- ularit y k of the set and, con trarily to the previous cases, it also dep ends on its b o x counting dimension D . Remark 2.5. We notic e that our pr o ofs also pr ovide estimates for the Kol- mo gor ov’s ε -entr opy of some families of Cantor sets [10] in the Hausdorff distanc e. 3 Pro ofs The follo wing t w o simple lemmas will b e us ed rep eatedly in the pro ofs her e- after. W e lea v e their elemen tary pr o of to the reader. Lemma 3.1. L et F and F ′ b e close d subsets of A . L et I = [ a, b ] and I ′ = [ a ′ , b ′ ] b e close d sub- intervals of A . L et H = [ c, d ] and H ′ = [ c ′ , d ′ ] b e close d subsets of ◦ I and ◦ I ′ r e sp e ctively. A ssu me that ∂ H ⊂ F , ∂ H ′ ⊂ F ′ , F ∩ ◦ H = ∅ and F ′ ∩ ◦ H ′ = ∅ . Mor e over assume that ther e exists ε > 0 such that | a − a ′ | ≤ ε , | b − b ′ | ≤ ε , | c − d | > 2 ε , | c ′ − d ′ | > 2 ε and max {| c − c ′ | , | d − d ′ |} > ε , then d H ( F , F ′ ) > ε . In the sequel, this lemma will b e used to sh o w that t wo Cantor sets ( F and F ′ ) are a Hausdorff distance larger than ε , H and H ′ pla ying the r ole of holes in the Can tor sets. 6 Lemma 3.2. L et (Ω , A , P ) b e a pr ob ability sp ac e. L et C b e a me asur able map fr om (Ω , A ) to the set of close d su b sets of A e quipp e d with the Bor el σ - algebr a induc e d by the Hausdorff metric. L et ( a k ) k b e a p ositive, incr e asing, diver ging se quenc e. Assume that for any i nte ger k ther e exists a se quenc e ( V k ,j ) 1 ≤ j ≤ 2 a k of me asur able subsets of Ω such that n ω : ∆( C ( ω ) , 2 − k ) < a k o ⊂ 2 a k [ j =1 V k ,j and X k 2 a k X j =1 P ( V k ,j ) < ∞ . Then, for P -almost every ω , ∆( C ( ω ) , 2 − k ) ≥ a k for any k lar ge enough (dep ending on ω ). 3.1 Pro of of T heorem 2.1 Let N b e the largest d egree of the p olynomial functions { φ i } , then we can write φ i ( x ) = X 0 ≤ α ≤ N c i,α x α ∀ i ∈ I with co efficien ts c i,α ∈ R . W e no w sho w how to construct a program P appro ximating C w ithin Hausdorff distance ε . Let ε b e fi xed and K a constan t to b e sp ecified later on. Let us defin e ε ′ = ε K . W e constru ct p olynomials ˜ φ i ( x ) := X 0 ≤ α ≤ N ˜ c i,α x α ∀ i ∈ I with co efficien ts s atisfying | c i,α − ˜ c i,α | < ε ′ ∀ i ∈ I ∀ 0 ≤ α ≤ N (10) suc h that the ˜ φ i ’s are inj ectiv e con tractions on A with uniform con traction rate ˜ ρ ∈ (0 , 1). F or any ω n 1 ∈ I n w e constru ct the comp osition ˜ φ ω n 1 as in (1). W e fir st sho w that f or any b oun ded set B ⊂ R suc h that φ i ( B ) ⊂ B for all i ∈ I with the same con traction r ate ρ , and ˜ φ i ( B ) ⊂ B for all i ∈ I , we ha ve for all n ∈ N d H ( φ ω n 1 ( B ) , ˜ φ ω n 1 ( B )) < ε ′ 1 − ρ n 1 − ρ sup x ∈ B       X 0 ≤ α ≤ N x α       ∀ ω n 1 ∈ I n . (11) 7 The pro of is b y indu ction. By (10) and definition of d H , one immediately gets d H ( φ i ( B ) , ˜ φ i ( B )) < ε ′ sup x ∈ B       X 0 ≤ α ≤ N x α       ∀ i ∈ I . The inductive step is ob tained b y using the triangle inequalit y for d H . By the first step we ha v e d H  φ ω n ( ˜ φ ω n − 1 1 ( B )) , ˜ φ ω n ( ˜ φ ω n − 1 1 ( B ))  < ε ′ sup x ∈ B       X 0 ≤ α ≤ N x α       where w e ha v e used ˜ φ ω n − 1 1 ( B ) ⊂ B . Moreo v er , by using the con traction rate ρ , we get d H  φ ω n ( φ ω n − 1 1 ( B )) , φ ω n ( ˜ φ ω n − 1 1 ( B ))  < ρ d H ( φ ω n − 1 1 ( B ) , ˜ φ ω n − 1 1 ( B )) < ε ′ ρ 1 − ρ n − 1 1 − ρ sup x ∈ B       X 0 ≤ α ≤ N x α       where the last inequalit y is the ( n − 1)-th step of the ind uction. Hence the triangle inequalit y imp lies that d H ( φ ω n 1 ( B ) , ˜ φ ω n 1 ( B )) < ε ′  1 + ρ 1 − ρ n − 1 1 − ρ  sup x ∈ B       X 0 ≤ α ≤ N x α       . This finish es the pr o of of (11 ). Let us c ho ose ¯ n ∈ N such that ρ ¯ n < ε ′ and ˜ ρ ¯ n < ε ′ . F or this fixed ¯ n , let V := { 0 , 1 } = ∂ [0 , 1] and define C := [ ω ¯ n 1 ∈I ¯ n ˜ φ ω ¯ n 1 ( V ) . W e no w prov e th at d H ( C , C ) < ε . Let us consider x ∈ C and y ∈ C . By (2) there exists z ∈ C su c h that x = φ ω ¯ n 1 ( x ) ( z ) for a giv en sequence ω ¯ n 1 ( x ) ∈ I n . Hence d ( x, y ) = d ( φ ω ¯ n 1 ( x ) ( z ) , y ) ≤ d H ( φ ω ¯ n 1 ( x ) ( z ) , φ ω ¯ n 1 ( x ) ( V )) + d H ( φ ω ¯ n 1 ( x ) ( V ) , ˜ φ ω ¯ n 1 ( x ) ( V )) + d H ( ˜ φ ω ¯ n 1 ( x ) ( V ) , y ) . (12) F or the first term w e u se the con traction prop erties to get d H ( φ ω ¯ n 1 ( x ) ( z ) , φ ω ¯ n 1 ( x ) ( V )) < ρ ¯ n diam ( A ) < ε ′ . 8 By (11), for the second term we ha v e d H ( φ ω ¯ n 1 ( x ) ( V ) , ˜ φ ω ¯ n 1 ( x ) ( V )) < ε ′ 1 − ρ ( N + 1) . If we tak e K = 1 + N + 1 1 − ρ then d ( x, y ) ≤ ε ′ K + d H ( ˜ φ ω ¯ n 1 ( x ) ( V ) , y ) = ε + d H ( ˜ φ ω ¯ n 1 ( x ) ( V ) , y ) . Cho osing y ∈ ˜ φ ω ¯ n 1 ( x ) ( V ) ⊂ C , we hav e d H ( ˜ φ ω ¯ n 1 ( x ) ( V ) , y ) = 0 , hence d ( x, y ) < ε . Therefore sup x ∈ C d ( x, C ) < ε. On the other hand f or a giv en ω ¯ n 1 ∈ I ¯ n and y ∈ ˜ φ ω ¯ n 1 ( V ), tak e x ∈ φ ω ¯ n 1 ( A ) ∩ C , noticing that this set is not empt y . Then w e d educe that sup y ∈ C d ( y , C ) < ε. Hence d H ( C , C ) < ε . Let us defin e the program P that con tains the num b ers ε , ρ and K , and su c h that it sp ecifies the co efficients { ˜ c i,α } , computes ¯ n and mak es the computation of the ˜ φ i ( V )’s. T he binary length ℓ ( P ) satisfies ℓ ( P ) 4 log  ( ε ′ ) − 1  = log K + log( ε − 1 ) . Indeed, ε is sp ecified with O (log ε − 1 ) bits, and ρ and K do n ot dep en d on ε and can b e appr o ximated by rational num b ers. The co efficien ts { ˜ c i,α } are appro ximations of the { c i,α } with precision ε ′ , h ence w e can c h o ose them as rational num b ers requir ing only O (log ( ε ′ ) − 1 ) bits of information. Finally the information for the computation of ¯ n and C needs only O (1) bits of information. Hence this pr ov es (3). W e now pro ve (4). Let I = { 0 , 1 } and define φ 1 ( x ) = bx , φ 2 ( x ) = 1 − bx f or b ∈ (0 , 1 / 2). W e denote by C b the Cantor set generated by the IFS { φ 1 , φ 2 } . T o b e in the con text of Lemma 3.2, we tak e b at random according to the uniform distribution on the interv al (1 / 4 , 1 / 3). W e restrict the p ossib le v alues of b to ensur e that the middle hole is large enough so that an ob viously simplified v ersion of Lemma 3.1 applies. 9 F or a fixed δ ∈ (0 , 1), d efine a k := (1 − δ ) k . F or an y k , there are at most 2 a k differen t binary programs ( P j ) 1 ≤ j ≤ 2 a k of length a k − 1, which generate at most 2 a k differen t sets C j := C ( P j ). W e d efine V k ,j := n b : d H ( C b , C j ) < 2 − k o . Then n b : ∆( C b , 2 − k ) < a k o ⊂ 2 a k [ j =1 V k ,j . W e no w estimate P ( V k ,j ). W e d enote b y ∂ + C j the right m ost p oint of C j ∩ (0 , 1 / 2). F or k large enough ( k ≥ 3) and for a given j , if b ∈ V k ,j then d ( b, ∂ + C j ) < 2 − k and therefore P ( V k ,j ) < 2 1 − k . Th is implies that X k 2 a k X j =1 P ( V k ,j ) < X k 2 a k 2 1 − k = X k 2 1 − δk < ∞ . The resu lt follo ws from Lemma 3.2. Remark 3.1. N otic e that this pr o of works also in arbitr ary finite dimension. 3.2 Pro of of T heorem 2.2 W e give the pro of in the case that the { φ i } are analytic functions on an op en ball B (0 , R ) ⊂ C of radiu s R > 1. T h e general case follo ws by applying the same argument to p iecewise p olynomial app ro ximations of the { φ i } . By hyp othesis we can write f or z ∈ A φ i ( z ) = ∞ X h =0 c i,h z h ∀ i ∈ I with co efficien ts c i,h ∈ C . By the analyticit y of the functions { φ i } in B (0 , R ) it follo ws that lim sup h →∞ | c i,h | R h ≤ 1 . Hence there exists a r eal constan t r suc h that | c i,h | R h ≤ r ∀ h ≥ 0 . As ab o ve let us denote b y V = { 0 , 1 } = ∂ [0 , 1]. W e no w construct app ro xi- mations of the analytic functions { φ i } . Let δ ∈ (0 , 1) b e such that δ R > 1, and , for ε fixed, d efine ε ′ := r ( δ R − 1) R (1 − δ ) ((1 − δ ) ε ) „ log R log 1 δ « . (13) 10 Let N b e the s mallest intege r satisfying ∞ X h = N δ h = δ N 1 − δ < (1 − ρ ) ε 4 r · (14) Hence we construct the p olynomials ˜ φ i ( z ) := N − 1 X h =0 ˜ c i,h z h ∀ i ∈ I with co efficien ts ˜ c i,h ∈ C suc h that | c i,h − ˜ c i,h | < ε ′ ∀ i ∈ I ∀ h = 0 , . . . , N − 1 (15) and the ˜ φ i ’s are cont ractions fun ctions on A with ˜ φ i ( A ) ⊂ A for all i ∈ I , and un iform con traction rate ˜ ρ ∈ (0 , 1). Let us choose ¯ n ∈ N suc h that ρ ¯ n < ε 4 R and ˜ ρ ¯ n < ε 4 R . F or this fixed ¯ n , w e define C := [ ω ¯ n 1 ∈I ¯ n ˜ φ ω ¯ n 1 ( V ) . W e now prov e that d H ( C , C ) < ε . T he p ro of will follo w again by using (12) and the analogue of (11). F or any b ound ed set B ⊂ B (0 , δ R ) su c h that φ i ( B ) ⊂ B and ˜ φ i ( B ) ⊂ B , w e ha ve for all n ∈ N d H ( φ ω n 1 ( B ) , ˜ φ ω n 1 ( B )) < ε 1 − ρ n 2 ∀ ω n 1 ∈ I n . (16) The pro of of (16) is by induction as th e pro of of (11). W e only s ho w the first step. By u sing (15) and (14), w e obtain for all z ∈ B (0 , δ R ) | φ i ( z ) − ˜ φ i ( z ) | =      N − 1 X h =0 ( c i,h − ˜ c i,h ) z h + ∞ X h = N c i,h z h      ≤ ε ′ N − 1 X h =0 δ h R h + r ∞ X h = N δ h < ε ′ δ N R N Rδ − 1 + (1 − ρ ) ε 4 r r < (1 − ρ ) ε   ε ′ R (1 − δ )((1 − δ ) ε )  − log R log δ − 1  4 r ( δ R − 1) + r 4 r   = (1 − ρ ) ε 2 , where in the last equalit y we hav e used (13). Hence d H ( φ i ( B ) , ˜ φ i ( B )) < ε (1 − ρ ) 2 ∀ i ∈ I . 11 The ind uctiv e step is obtained as in the p ro of of (11) by the triangle in- equalit y for d H . W e no w write (12) and, by r ep eating the argument of the p ro of of T he- orem 2.1 and by (16), we obtain d H ( C , C ) < ε . Let us define the program P wh ic h con tains the n u m b ers ε , R , r , δ , N and ¯ n , and such that it sp ecifies the co efficients { ˜ c i,h } for h = 0 , . . . , N − 1, and mak es the computation of the ˜ φ i ( V )’s. T he binary length ℓ ( P ) satisfies ℓ ( P ) 4 N log  ( ε ′ ) − 1  4  log( ε − 1 )  2 . Indeed, ε is sp ecified with O (log ε − 1 ) bits, and ¯ n ≍ log ε − 1 . Hence it is sp ecified b y o (log ε − 1 ) bits and N ≍ log ε − 1 , hence it is sp ecified by o (log ε − 1 ) bits. The coefficients { ˜ c i,h } are appro ximations of the { c i,h } with precision ε ′ , hence w e can choose them as rational n u m b ers requ iring only O (log( ε ′ ) − 1 ) b its of inf orm ation, but there are N of them for eac h fu nction φ i (see (13) and (14)), h ence we n eed O ( N log ( ε ′ ) − 1 ) bits of information, that is O (  log( ε − 1 )  2 ). Finally R , r and δ d o not d ep end on ε , hence the information for them and for the compu tation of C need only O (1) bits of information. Hence ∆( C , ε ) 4  log( ε − 1 )  2 and the theorem is p ro ved. 3.3 Pro of of T heorem 2.3 W e first p ro ve (6) by constructing an appro xim ation C of the set C . Let us consider a fixed sequence λ ∈ (0 , 1) N and the Cant or s et C λ . F or a fixed ε , let ¯ n b e giv en b y ¯ n := min  n ∈ N : 1 2 n < ε 2  . (17) Next, let u s consider a sequ ence ˜ λ = ( ˜ λ k ) k ∈ N ∈ (0 , 1) N suc h that | λ 1 − ˜ λ 1 | < ε and | λ k − ˜ λ k | < 2 k − 2 ε ∀ k = 2 , . . . , ¯ n . (18) Then we define the appro ximation C of C λ to b e the fi nite set C := ∂ C ¯ n ˜ λ = 2 ¯ n [ i =1 ∂ J ¯ n i ( ˜ λ ) where the sets C ¯ n ˜ λ and J ¯ n i ( ˜ λ ) are constructed as sp ecified in S ection 2. W e no w pro ve that d H ( C λ , C ) < ε . T o this aim we sho w that d H ( ∂ J k 1 ( λ ) , ∂ J k 1 ( ˜ λ )) < ε 2 (19) 12 for all k = 1 , . . . , ¯ n . The same argument applies to all other sets J k i . This is enough since it implies that, for any tw o p oin ts x ∈ C λ and y ∈ C in the analogous int erv als ( i.e. , x ∈ J ¯ n i ( λ ) and y ∈ ∂ J ¯ n i ( ˜ λ ) with the same ind ex i = 1 , . . . , 2 ¯ n ), d ( x, y ) ≤ d ( x, ∂ J ¯ n i ( λ )) + d H ( ∂ J ¯ n i ( λ ) , ∂ J ¯ n i ( ˜ λ )) + d ( ∂ J ¯ n i ( ˜ λ ) , y ) < 1 2 ¯ n + ε 2 + 0 < ε, where for the first term w e ha v e u sed (5) and (17), for the second term we ha ve used (19), and for the third term we ha ve us ed the definition of C . Hence d H ( C λ , C ) < ε . It remains to p ro ve (19). By definition ∂ J k 1 ( λ ) = ( 0 , 1 2 k k Y h =1 λ h ) , ∂ J k 1 ( ˜ λ ) = ( 0 , 1 2 k k Y h =1 ˜ λ h ) . Hence it is enough to show that d 1 2 k k Y h =1 λ h , 1 2 k k Y h =1 ˜ λ h ! < ε 2 (20) for all k = 1 , . . . , ¯ n . By definition of ˜ λ , it h olds d  1 2 λ 1 , 1 2 ˜ λ 1  < ε 2 . Assuming that (20) holds for k − 1, one gets d 1 2 k k Y h =1 λ h , 1 2 k k Y h =1 ˜ λ h ! ≤      1 2 k − 1 k − 1 Y h =1 λ h           λ k − ˜ λ k 2      + d 1 2 k − 1 k − 1 Y h =1 λ h , 1 2 k − 1 k − 1 Y h =1 ˜ λ h ! ˜ λ k 2 < 1 2 k − 1 2 k − 2 ε 2 + ε 4 < ε 2 , where w e ha v e used (18) for the first term, the inductiv e h yp othesis for the second term, and the fact that λ, ˜ λ ∈ (0 , 1) N . T o finish the pro of of (6) w e ha ve to estimate the length of a binary program P pro du cing the finite set C . The program P must cont ain th e n u m b er ε , the information to compute ¯ n , the con traction factors ( ˜ λ k ) for k = 1 , . . . , ¯ n and the instruction to compu te C . The n u m b er of instructions for all the computations are O (1) with resp ect to ε . Th e num b er ε is sp ecified b y O (log ε − 1 ) bits of information and ¯ n ≈ log ε − 1 and eac h co efficien t ˜ λ k 13 needs O (log ε − 1 ) b its of information. Since there are ¯ n co efficien ts to b e sp ecified, we find ℓ ( P ) 4  log( ε − 1 )  2 hence (6) follo ws. W e no w prov e (7). First of all w e iden tify a f u ll measure set of “go o d” λ . By th e hypothesis on th e density f ( x ) of the common distribu tion of the random v ariables ( λ k ), the f ollo wing quant ity is finite γ := Z 1 0 log( x ) f ( x ) dx < 0 . Notice that e γ 2 can b e interpreted as the t yp ical con traction rate, since p ro d- ucts of man y i.i.d. r andom v ariables λ k will b e inv olv ed. Giv en an y η > 0 w e define, for all n ∈ N , Λ n := ( λ ∈ (0 , 1) N : k Y h =1 λ h > e k ( γ − η ) , ∀ k = ⌊ √ n ⌋ , . . . , n ) . (21) W e remark that, since λ k ∈ (0 , 1) for all k ≥ 1, if λ ∈ Λ n then k Y h =1 λ h > e √ n ( γ − η ) ∀ k = 1 , . . . , ⌊ √ n ⌋ − 1 . (22) Lemma 3.3. L et us denote by Λ c n the c omplement of Λ n in (0 , 1) N , then X n P (Λ c n ) < ∞ , i.e. , almost e v ery λ ∈ (0 , 1) N b elongs to Λ c n only for finitely many n ∈ N . Pr o of. W e us e the large d eviation pr inciple for indep end en t and identic ally distributed rand om v ariables (see, e.g. , [8]). It implies that for an y fixed η > 0 there exists a p ositive constan t C s uc h that lim k →∞ 1 k log P  log( λ 1 × · · · × λ k ) k − γ < − η  = − C . Hence, for n large enough and for all 0 < C ′ < C , w e ha ve the estimate P (Λ c n ) ≤ n X k = ⌊ √ n ⌋ P ( k Y h =1 λ h < e k ( γ − η ) ) ≤ e − C ′ √ n n X k = ⌊ √ n ⌋ e − C ′ ( k − √ n ) . Therefore P (Λ c n ) 4 e − C ′ √ n and the lemma follo ws by the Borel-Can telli Lemma. 14 F or an y ε w e d efine N ( ε ) := min n n ∈ N :  2 e η − γ  n ε >  log( ε − 1 )  − 2 o . (23) W e no w consider a sub s et of Λ N ( ε ) . Let q ∈ N an d define Ψ q := ( λ ∈ Λ N (2 − q ) : (1 − λ k +1 ) Q k h =1 λ h 2 k > 2 2 − q , ∀ k = 0 , . . . , N (2 − q ) ) . Lemma 3.4. We have X q P (Ψ c q ) < ∞ i.e. , almost e v ery λ ∈ (0 , 1) N lies in Ψ c q only for finitely many q ∈ N . Pro of. By Lemm a 3.3 and the Borel-Cant elli Lemma, it is enough to pro ve that X q P  Ψ c q ∩ Λ N (2 − q )  < ∞ . First obser ve that if 0 ≤ k ≤ p N (2 − q ) and (1 − λ k +1 ) > 2 2 − q 2 k ( e η − γ ) √ N (2 − q ) then λ satisfies (1 − λ k +1 ) Q k h =1 λ h 2 k > 2 2 − q . (24) Similarly , if p N (2 − q ) ≤ k ≤ N (2 − q ) and (1 − λ k +1 ) > 2 2 − q 2 k ( e η − γ ) k , then (24) holds. Ther efore P  λ ∈ Λ N (2 − q ) \ Ψ q  ≤ 2 3 − q 2 √ N (2 − q ) e ( η − γ ) √ N (2 − q ) + + 2 2 − q 2 N (2 − q )+1 e ( η − γ )( N (2 − q )+1) ≤ O (1) q 2 whic h is sum m able o ve r q . Th e lemma is pr o v ed. F or a given ε , let λ ∈ Ψ log( ε − 1 ) and defi ne M ε,N ( ε ) ( λ ) := Ψ log( ε − 1 ) \        ˜ λ : | λ 1 − ˜ λ 1 | < 2 ε | λ k − ˜ λ k | < 2 k +1 ( e η − γ ) √ N ε if k = 2 , . . . , ⌊ √ N ⌋ − 1 | λ k − ˜ λ k | < 2 e η − γ (2 e η − γ ) k ε if k = ⌊ √ N ⌋ , . . . , N        (25) 15 W e no w show that if ˜ λ 6∈ M ( λ ) th en d H ( C λ , C ˜ λ ) ≥ ε . This f ollo ws f rom Lemma 3.1 and we n o w c heck the hyp othesis to b e satisfied. If ˜ λ ∈ Ψ log( ε − 1 ) \ M ( λ ) then one of the conditions in (25) is violated. F ollo win g the n otatio n of Lemma 3.1, w e start with I = I ′ = [0 , 1]. W e tak e H = [ λ 1 2 , 1 − λ 1 2 ] and H ′ = [ ˜ λ 1 2 , 1 − ˜ λ 1 2 ]. Then | H | > 2 ε and | H ′ | > 2 ε sin ce λ and ˜ λ are in Ψ log( ε − 1 ) . If | λ 1 − ˜ λ 1 | ≥ 2 ε , then    λ 1 2 − ˜ λ 1 2    ≥ ε , and Lemma 3.1 applies with F = C λ and F ′ = C ˜ λ , implying d H ( C λ , C ˜ λ ) > ε . Assume that, for some k = 2 , . . . , ⌊ √ N ⌋ − 1, all conditions in (25) are satisfied up to k − 1 and condition k is violate d . Either there is an ℓ < k suc h that      1 2 ℓ ℓ Y h =1 λ h − 1 2 ℓ ℓ Y h =1 ˜ λ h      > ε, in wh ich case we define ˆ k to b e the smallest such ℓ . Or , if no suc h ℓ exists,      1 2 k k Y h =1 λ h − 1 2 k k Y h =1 ˜ λ h      ≥    λ k − ˜ λ k    2 k k − 1 Y h =1 λ h − ˜ λ k 2 k      k − 1 Y h =1 λ h − k − 1 Y h =1 ˜ λ h      ≥ ≥ 2 ε  e η − γ  √ N k − 1 Y h =1 λ h − ε > ε where w e ha v e used (22) an d the fact that the leftmost p ositiv e p oints u p to the ( k − 1)-th step of the construction are ε -close, and w e set ˆ k = k . W e will app ly Lemma 3.1 with I = J ˆ k − 1 1 ( λ ) and I ′ = J ˆ k − 1 1 ( ˜ λ ). W e tak e H =   1 2 ˆ k ˆ k Y h =1 λ h ,  1 − λ ˆ k 2  1 2 ˆ k − 1 ˆ k − 1 Y h =1 λ h   and H ′ =   1 2 ˆ k ˆ k Y h =1 ˜ λ h , 1 − ˜ λ ˆ k 2 ! 1 2 ˆ k − 1 ˆ k − 1 Y h =1 ˜ λ h   . W e hav e | H | > 2 ε and | H ′ | > 2 ε since λ and ˜ λ are in Ψ log( ε − 1 ) , and we can apply Lemma 3.1 which giv es d H ( C λ , C ˜ λ ) > ε . The same argumen t applies if the k -th condition with k = √ N , . . . , N is violated, and all conditions up to k − 1 are satisfied. If the leftmost p ositiv e p oint s up to the ( k − 1)-th step of the construction are ε -close, we get      1 2 k k Y h =1 λ h − 1 2 k k Y h =1 ˜ λ h      ≥ 2 ε  e η − γ  k − 1 k − 1 Y h =1 λ h − ε > ε b y defin ition (21) of Λ N ( ε ) and u sing, as ab o ve, that all p revious leftmost p os- itiv e p oin ts are ε -close to eac h other. Again this implies that d H ( C λ , C ˜ λ ) > ε . 16 Let us now estimate the measure of the set M ε,N ( ε ) ( λ ). By an easy computation based on the ind ep endence of the rand om v ariables ( ˜ λ k ), w e obtain that for all λ ∈ Ψ log( ε − 1 ) P ( M ε,N ( ε ) ( λ )) ≤  max x ∈ [0 , 1] f ( x )  N ( ε ) (2 ε ) N ( ε ) ( e η − γ ) − √ N ( ε ) 2 P N ( ε ) k =2 k ( e η − γ ) P N ( ε ) k = ⌊ √ N ( ε ) ⌋ k ≤  2 e η − γ  − N ( ε ) 2 2 + O ( N ( ε ) ) = e − O (1)(log( ε − 1 )) 2 , (26) where we ha ve used the definition (23) of N ( ε ). Note th at this estimate is uniform in λ ∈ Ψ log( ε − 1 ) . F or a fixed δ ∈ (0 , 1), defin e a q := q 2 − δ . F or any q , there are at m ost 2 a q differen t binary programs ( P j ) 1 ≤ j ≤ 2 a q of length a q − 1, whic h generate at most 2 a q differen t sets C j := C ( P j ). W e define V q ,j :=  λ : d H ( C λ , C j ) < 2 − q  . Then  λ : ∆( C λ , 2 − q ) < a q  ⊂ 2 a q [ j =1 V q ,j . W e can w r ite P   2 a q [ j =1 V q ,j   ≤ P (Λ c N (2 − q ) ) + P (Ψ c q ∩ Λ N (2 − q ) ) + 2 a q X j =1 P ( V q ,j ∩ Ψ q ) . Moreo v er if V q ,j ∩ Ψ q 6 = ∅ , there is a λ ∈ Ψ q suc h that V q ,j ∩ Ψ q ⊂ M 2 − q ,N (2 − q ) ( λ ). By Lemmas 3.3 and 3.4, and by (26 ) it follo ws that X q 2 a q X j =1 P ( V q ,j ) < ∞ . The resu lt follo ws from Lemma 3.2. 3.4 Pro of of T heorem 2.4 Preliminaries. W e first recall th e definition of the sc aling fu nction S C of a C k Can tor set C ([14],[13]). In the sequel we fix I = { 0 , 1 } . F or a w ord ω n 1 ∈ I n w e let J ω n 1 := φ ω n 1 ([0 , 1]) = φ ω n ◦ φ ω n − 1 ◦ · · · ◦ φ ω 1 ([0 , 1]) . Then by definition φ i ( J ω n 1 ) = J ω n 1 i for any i ∈ I , and it holds J ω n 1 ⊂ J ω n 2 ⊂ · · · ⊂ J ω n n − 1 ⊂ J ω n . 17 The scaling function describ es the con traction rates in the previous inclu- sions. F or a wo r d ω n 1 ∈ I n w e define ˜ S C ( ω n 1 ) ∈ (0 , 1) 2 . The tw o comp onen ts of ˜ S C ( ω n 1 ) are th e rates of con tractions ( ˜ S C ( ω n 1 )) i = | J iω n 1 | | J ω n 1 | i = 0 , 1 where | J | d enotes the length of the in terv al J . The length of the gap b et ween the t wo inte r v als J 0 ω n 1 and J 1 ω n 1 in J ω n 1 can b e r econstructed f rom these data. The scaling function is defined to b e the function S C : I ∞ → (0 , 1) 2 giv en b y S C ( ω ) := lim n →∞ ˜ S C ( ω n 1 ) . W e refer to [14] for the p r o of of the existence of this limit. By d efi nition one has | J ω n 1 | = n − 1 Y j =1  ˜ S C ( ω n j +1 )  ω j . By u sing the scaling function w e can introd uce a d istance d S ( ω , ˜ ω ) on I ∞ in the f ollo wing wa y . F or t wo s equences ω , ˜ ω ∈ I ∞ , let ω ∩ ˜ ω denote their longest common prefix, and let | ω ∩ ˜ ω | denote its length. Then we let d S ( ω , ˜ ω ) := sup α ∈ I ∞ n = | ω ∩ ˜ ω | Y j =1  S C ( ω n j +1 α )  ω j . (27) Then there exists a constan t H > 0 such that for any ω , ˜ ω ∈ I ∞ it holds 1 H ≤ J ω ∩ ˜ ω d S ( ω , ˜ ω ) ≤ H . Relations b et ween the prop erties of the scaling function of a C k cen tral Can tor set and th e different iability of the IFS generating this Canto r set ha ve b een s tudied in [13] in the case k ≥ 1 (we refer the r eader to Main Theorem [13], p age 406). The idea is the follo wing. Let A ( ω n 1 ) denote the set of the fou r b oundary p oint s of the int erv als ( J iω n 1 ) i ∈I . A scaling function S C generates a C k Can tor set C if and only if for an y n ∈ N there are diffeomorph isms f rom A ( ω n 1 ) into A ( ˜ ω n 1 ), for an y ω n 1 6 = ˜ ω n 1 ∈ I n , with deriv ativ es b ound ed by a constant C ( ω n 1 , ˜ ω n 1 ) wh ic h satisfies C ( ω n 1 , ˜ ω n 1 ) = C d S ( ω n 1 α, ˜ ω n 1 α ) k − 1 (28) where C do es not d ep end on n and from the d efinition of d S the righ t hand side is indep end en t on α ∈ I ∞ . 18 Pro of of Theorem 2.4. W e fi rst pro ve (8). Let ε b e fixed. W e sho w ho w to app ro ximate th e set C w ith in Haus d orff distance ε . W e will give th e pro of for in teger k ≥ 1. Th e pro of easily extends to functions whose k -th deriv ativ e is H¨ older. W e can w r ite the T aylo r expansions of the maps φ i at a p oin t x 0 φ i ( x ) = k − 1 X p =0 c i,p ( x 0 ) ( x − x 0 ) p + R i ( x, x 0 ) ∀ x ∈ [0 , 1] . Moreo v er there exists a constant K > 0 su c h that | R i ( x, x 0 ) | ≤ K | x − x 0 | k for x ∈ [0 , 1], for all i ∈ I and x 0 ∈ [0 , 1]. Let ε ′ = ε 1 − ρ M for a constant M to b e sp ecified later on. W e no w construct a sequence of p olynomials which approximat e the maps ( φ i ) i ∈I . If D is the b o x counting dimension of the Can tor set C , we need for an y δ > 0 at most N = O (( ε ′ ) − D k − δ ) in terv als ( I s ) s =1 ,...,N of size ( ε ′ ) 1 k to co v er C . Hence w e can consider the maps ( φ i ) i ∈I restricted to the sets ( I s ). If y s denotes th e mid dle p oin t of the interv al I s , let ˜ y s b e th e app ro ximation of the p oint y s within a distance ε ′ . Then w e define ˜ φ s i ( x ) = k X p =0 ˜ c i,p ( y s ) ( x − ˜ y s ) p ∀ x ∈ [0 , 1] suc h that | c i,p ( y s ) − ˜ c i,p ( y s ) | < ε ′ ∀ i ∈ I ∀ p = 0 , . . . , k (29) and they are contrac tions on R with the s ame u n iform con traction rate ˜ ρ < 1. T o construct an appro ximation of C , w e w ork on the b oundary p oin ts of the interv als J ω n 1 whic h all are in C . Let u s denote J ω n 1 = [ y 1 ω n 1 , y 2 ω n 1 ]. Since for an y n ∈ N and any ω n 1 ∈ I n w e hav e y η ω n 1 ∈ C for η = 1 , 2, we can asso ciate to a give n y η ω n 1 a sequence σ n − 1 0 ∈ { 1 , . . . , N } n − 1 whic h sp ecifies to which in terv als of the co ver ( I s ) s =1 ,...,N the pre-images y η ω n − 1 1 , y η ω n − 2 1 , . . . , y η ω 1 , y η ♯ of y η ω n 1 b elong, where y η ♯ ∈ { 0 , 1 } . W e no w establish the analogue of (11) for the b oun dary p oin ts. Let us define ˜ y η ω n 1 := ˜ φ σ n − 1 ω n ◦ ˜ φ σ n − 2 ω n − 1 ◦ · · · ◦ ˜ φ σ 0 ω 1 ( y η ♯ ) then for all n ∈ N it holds | y η ω n 1 − ˜ y η ω n 1 | <   k + 1 + K + max i ∈I , s =1 ,.. .,N k X p =0 | c i,p ( y s ) |   ε ′ 1 − ρ ∀ ω n 1 ∈ I n . (30) 19 The pro of is b y induction. T he first step ( n = 1) follo ws b y definition of the appro ximating p olynomials, estimates (29 ) and pr op erties of the remaind er R i ( x, x 0 ). Th is yields | y η i − ˜ y η i | = | φ i ( y η ♯ ) − ˜ φ σ 0 i ( y η ♯ ) | ≤ ε ′ k X p =0 ( ε ′ ) p k + k X p =0 | c i,p ( y σ 0 ) || ( y η ♯ − y σ 0 ) p − ( y η ♯ − ˜ y σ 0 ) p | + K ε ′ ≤   k + 1 + K + max i ∈I , s =1 ,.. .,N k X p =0 | c i,p ( y s ) |   ε ′ . The ind uctiv e step follo ws by using the triangle inequality | y η ω n 1 − ˜ y η ω n 1 | ≤ | φ ω n ( y η ω n − 1 1 ) − ˜ φ σ n − 1 ω n ( y η ω n − 1 1 ) | + | ˜ φ σ n − 1 ω n ( y η ω n − 1 1 ) − ˜ φ σ n − 1 ω n ( ˜ y η ω n − 1 1 ) | together with | φ ω n ( y η ω n − 1 1 ) − ˜ φ σ n − 1 ω n ( y η ω n − 1 1 ) | < ε ′   k + 1 + K + max i ∈I , s = 1 , ...,N k X p =0 | c i,p ( y s ) |   and | ˜ φ σ n − 1 ω n ( y η ω n − 1 1 ) − ˜ φ σ n − 1 ω n ( ˜ y η ω n − 1 1 ) | < ρ | y η ω n − 1 1 − ˜ y η ω n − 1 1 | , where ρ is the uniform con traction r ate of the approxi mating p olynomials. Let us c ho ose ¯ n such that ρ ¯ n < ε 2 for all n ≥ ¯ n . Then we define the set C := [ ω ¯ n 1 ∈I ¯ n  ˜ y 1 ω ¯ n 1 ∪ ˜ y 2 ω ¯ n 1  and we claim that d H ( C , C ) < ε . Indeed, b y defin ition of ¯ n , an y p oin t in the Can tor set C is at most at distance ε 2 from a p oint in the b oundary of one of the s ets J ω ¯ n 1 . Moreo ver, b y construction of the p oin ts ˜ y η ω ¯ n 1 w e hav e (30), hence the claim follo ws sin ce d H ( C , C ) ≤ sup x ∈ C inf ω ¯ n 1 ∈I ¯ n | x − ˜ y η ω ¯ n 1 | ≤ sup x ∈ C inf ω ¯ n 1 ∈I ¯ n  | x − y η ω ¯ n 1 | + | y η ω ¯ n 1 − ˜ y η ω ¯ n 1 |  < ε pro vid ed that w e c ho ose M := 2   k + 1 + K + max i ∈I , s = 1 , ...,N k X p =0 | c i,p ( y s ) |   . 20 Let us d efine the program P that con tains the num b ers ε , ρ , D , M , K and k , and su c h th at it sp ecifies all the necessary co efficien ts ˜ c i,p , makes the computation to obtain N and the approximat ed p oin ts ˜ y s , and moreo ve r it mak es the computations to obtain ¯ n and the p oint s ˜ y η ω ¯ n 1 . Th e bin ary length ℓ ( P ) satisfies ℓ ( P ) 4 ε − D k − δ since ε is sp ecified with O (log( ε − 1 )) bits; ρ , D , M , K and k do n ot dep end on ε and can b e app ro ximated by rational num b ers . The co efficien ts ˜ c i,p and the p oin ts ˜ y s are constru cted as in the previous p ro ofs with p recision ε ′ , hence eac h of them needs O (log( ε − 1 )) bits of in formation and their num b er is N ( k + 2) = O ( ε − D k − δ ). Finally all the computations to obtain C need O (1) bits of instru ctions. Hence (8) follo ws. W e no w pro ve (9). W e d efine a class of particular scaling fun ctions S ( α ) on I ∞ to construct d ifferen tiable C an tor sets with the give n distortion complexit y . Let us denote by I ∗ := ∪ n ∈ N I n the countable set of fin ite strings s written us in g the alphab et I . Let ( λ s ) s ∈I ∗ b e a family of indep endent iden tically distributed random v ariables with v alues in the in terv al (0 , 1) and absolutely contin uous distribution with densit y f ( x ) b oun d ed ab o ve and b elo w aw a y fr om zero. Note that the empt y string ♯ b elongs to I ∗ and therefore ther e is an asso ciated r an d om v ariable λ ♯ . Let 0 < ζ < 1, 0 < ρ < 1 and ρ < θ < 1 b e giv en constan ts, with ρ determining the contract ion r ate. ζ will b e chosen small enough later on . W e will only consider central Can tor sets, namely the tw o comp onents of the scaling functions will b e equal. W e d efine the scaling fun ction S λ ( α ) := ρ + ζ ∞ X q =1 θ q − 1 λ α q 1 ∀ α ∈ I ∞ whic h d ep ends on the realisation of th e family ( λ s ). W e remark that for an y realisation it holds ρ ≤ S λ ( α ) ≤ ρ + ζ 1 − θ ∀ α ∈ I ∞ . (31) Hence if ζ is small enough, th e rate of contrac tion is almost ρ . It is also useful to define the truncated scaling function ˜ S λ b y ˜ S λ ( ω m 1 ) := ρ + ζ m X q =1 θ q − 1 λ ω q 1 ∀ m ≥ 0 . (32) Using the relations | J iω m 1 | | J ω m 1 | = ˜ S λ ( ω m 1 ) ∀ i ∈ I (33) 21 w e can construct a cen tral Cantor set C λ generated b y the scaling fu nction S λ . F rom (31) it follo ws that the Cantor set C λ has b o x coun ting dimension D ( ζ ) whic h satisfies D ( ζ ) = − log 2 log ρ + O ( ζ ) as ζ → 0 . (34) W e now consider the differen tiabilit y of the IFS generating C λ . By (31) and the definition (27) of d S it follo ws ρ n ≤ d S ( ω , ˜ ω ) ≤  ρ + ζ 1 − θ  n n = | ω ∩ ˜ ω | for any ω , ˜ ω ∈ I ∞ . Moreo ver for an y ω 6 = ˜ ω it h olds | S λ ( ω ) − S λ ( ˜ ω ) | ≤ 2 ζ θ n +1 1 − θ n = | ω ∩ ˜ ω | . Hence for m > n we ha ve diffeomorphisms f rom A ( ω m 1 ) into A ( ˜ ω m 1 ) with deriv ativ es b ounded by a constant C ( ω m 1 , ˜ ω m 1 ) = O ( θ n +1 ). Th ese facts to- gether with relation (28) imply th at the Cantor set C λ is of class C k with k = 1 + log θ log ρ ≥ 1 . (35) Let 0 < ε < 1 b e fixed and small en ough dep end in g on the constant s ρ, θ , ζ . Let λ := ( λ s ) and λ ′ := ( λ ′ s ) denote tw o differen t realisations of the family of random v ariables. W e give a condition on λ and λ ′ to h a ve d H ( C λ , C λ ′ ) > ε . W e denote ¯ p =  log( C ε ζ − 1 ) log( ρθ )  (36) where C is a p ositiv e constant (indep enden t of ε ) to b e sp ecified later on. F or an y σ ∈ I ∗ , w e d enote b y J σ and J ′ σ the int er v als asso ciated to σ in C λ and C λ ′ resp ectiv ely . Lemma 3.5. Assume ther e is 0 ≤ p ≤ ¯ p satisfying max ω p 1 ∈I p d H ( J ω p 1 , J ′ ω p 1 ) > ε. (37) Then d H ( C λ , C λ ′ ) > ε . Pr o of. Denote by p th e smallest in teger for whic h th e ab o ve inequ alit y holds and b y ω p 1 ∈ I p the string realising the maximum. W e apply Lemma 3.1 with I = J ω p − 1 1 and I ′ = J ′ ω p − 1 1 . The hyp otheses on I and I ′ follo w by the fact that (37) is violated up to p − 1. Th e gaps H and H ′ ha ve size at least ρ p − 1 > ( ρθ ) ¯ p = C ε ζ − 1 > 2 ε 22 for ε small enough if C ζ − 1 > 2. Finally since d H ( J ω p 1 , J ′ ω p 1 ) > ε w e hav e all th e hyp otheses of Lemma 3.1. Hence the lemma follo w s . Lemma 3.6. A ssume that ther e exists 0 ≤ p ≤ ¯ p and a se q u enc e ω p 1 ∈ I p such that | λ ω p 1 − λ ′ ω p 1 | > ( ρθ ) − p (4 + 2 ρ 2 ) ε ζ · Then d H ( C λ , C λ ′ ) > ε . Pr o of. Denote b y p the smallest in teger for whic h the ab o v e in equalit y holds. It is enough to assum e that for an y q < p condition (37) is not v erified, otherwise the pro of follo ws immediately fr om Lemma 3.5. Let x (resp ectiv ely x ′ ) b e the p oin t in the b ound ary of J ω p 1 (resp ectiv ely J ′ ω p 1 ) wh ic h is not in the b ound ary of J ω p 2 (resp ectiv ely J ′ ω p 2 ). Let y (resp ec- tiv ely y ′ ) b e the other b oun d ary p oint of J ω p 2 (resp ectiv ely J ′ ω p 2 ). Since by the recursive assumption | y − y ′ | ≤ ε , we ha v e | x − x ′ | ≥ | x − y − ( x ′ − y ′ ) | − ε. No w since | x − y | = | J ω p 1 | and ( x − y ) and ( x ′ − y ′ ) ha ve the same sign, we get | x − x ′ | ≥   | J ω p 1 | − | J ′ ω p 1 |   − ε This can also b e written | x − x ′ | ≥   ˜ S λ ( ω p 2 ) | J ω p 2 | − ˜ S λ ′ ( ω p 2 ) | J ′ ω p 2 |   − ε ≥ ≥   ˜ S λ ( ω p 2 ) − ˜ S λ ′ ( ω p 2 )   | J ω p 2 | + | J ′ ω p 2 | 2 − ˜ S λ ( ω p 2 ) + ˜ S λ ′ ( ω p 2 ) 2   | J ω p 2 | − | J ′ ω p 2 |   − ε ≥ ≥   ˜ S λ ( ω p 2 ) − ˜ S λ ′ ( ω p 2 )   | J ω p 2 | + | J ′ ω p 2 | 2 − 3 ε ≥ ρ p   ˜ S λ ( ω p 2 ) − ˜ S λ ′ ( ω p 2 )   − 3 ε since d H  J ω p 2 , J ′ ω p 2  ≤ ε and S λ ≥ ρ . F rom (32) we get ˜ S λ ( ω p 2 ) − ˜ S λ ′ ( ω p 2 ) = ζ p X q =2 θ q − 2  λ ω q 2 − λ ′ ω q 2  and fr om (33) we ha ve ζ p − 1 X q =2 θ q − 2  λ ω q 2 − λ ′ ω q 2  = ( | J ω p − 1 1 | − | J ′ ω p − 1 1 | )( | J ω p − 1 2 | + | J ′ ω p − 1 2 | ) 2 | J ω p − 1 2 || J ′ ω p − 1 2 | + ( | J ω p − 1 1 | + | J ′ ω p − 1 1 | )( | J ′ ω p − 1 2 | − | J ω p − 1 2 | ) 2 | J ω p − 1 2 || J ′ ω p − 1 2 | · 23 Since | J ω p − 1 1 | ≤ | J ω p − 1 2 | ≤ ( ρ + ζ 1 − θ ) p − 2 w e get (using again d H  J ω p 2 , J ′ ω p 2  ≤ ε , and S λ ≥ ρ )       ζ p − 1 X q =2 θ q − 2  λ ω q 2 − λ ′ ω q 2        ≤ 2 ρ − p +2 ε . Hence | ˜ S λ ( ω p 2 ) − ˜ S λ ′ ( ω p 2 ) | ≥ ζ θ p − 2 | λ ω p 1 − λ ′ ω p 1 | − 2 ρ − p +2 ε. W e conclude th at | x − x ′ | ≥ ζ ( ρθ ) p | λ ω p 1 − λ ′ ω p 1 | − (3 + 2 ρ 2 ) ε. Therefore the lemma follo w s by applying Lemma 3.1. W e n o w w ant to estimate for a give n realisation λ of the family ( λ s ) the probabilit y of the ev ent E ¯ p ( λ ) = sup 0 ≤ q ≤ ¯ p ( ρθ ) q sup σ ∈I q | λ σ − λ ′ σ | ≤ (4 + 2 ρ 2 ) ε ζ · By ind ep endence of the family ( λ s ) w e get P ( E ¯ p ( λ )) ≤ ¯ p Y q =0  C ( ρθ ) − q ε ζ − 1  2 q where C = (4 + 2 ρ 2 ) sup x ∈ [0 , 1] f ( x ) where f is the d en sit y of the random v ariables ( λ s ). These relations imply th at log P ( E ¯ p ( λ )) ≤ ¯ p X q =0 2 q  log( C ε ζ − 1 ) − q log ( ρθ )  ≤ 2 ¯ p +1 log( ρθ ) − (3 + ¯ p ) log ( ρθ ) . F or a fixed δ ∈ (0 , 1), defin e a ℓ := 2 ℓ ( − δ + D k ) and choose ε = 2 − ℓ . F or any ℓ , there are at m ost 2 a ℓ differen t binary p rograms ( P j ) 1 ≤ j ≤ 2 a ℓ of length a ℓ − 1, whic h generate at most 2 a ℓ differen t sets C j := C ( P j ). W e d efine V ℓ,j := n λ : d H ( C λ , C j ) < 2 − ℓ o . 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