On Fractal Properties of Weierstrass-Type Functions

Proceedings of the International Geometry Center Vol. 12, no. 2 (2019) pp. 43–61

On fractal properties of Weierstrass-type functions

Claire David

Abstract. In the sequel, starting from the classical Weierstrass function + 8 n n deﬁned, for any real number x, by (x) = λ cos (2 π Nb x), where λ W n=0 ÿ and Nb are two real numbers such that 0 λ 1, Nb N and λ Nb 1, we highlight intrinsic properties of curious mapsă ă which happenP to constituteą a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-diﬀerentiabilty of Weierstrass type functions.

Анотація. Метою даної роботи є узагальнення попередніх результатів автора про класичну функцію Вейерштрасса та її графік. Його можна отримати як границю послідовності префракталів, тобто графів, отри- маних за допомогою ітераційної системи функцій, які, як правило, не є стискаючими відображеннями. Натомість вони мають в деякому сенсі еквівалентну властивість до стискаючих відображень, оскільки на ко- жному етапі ітераційного процесу, який дає змогу отримати префра- ктали, вони зменшують двовимірні міри Лебега заданої послідовності прямокутників, що покривають криву. Такі системи функцій відіграють певну роль на першому кроці процесу побудови підкови Смейла. Вони можуть бути використані для доведення недиференційованості функції Вейєрштрасса та обчислення box-розмірності її графіка, а також для по- будови більш широких класів неперервних, але ніде не диференційовних функцій. Останнє питання ми вивчатимемо в подальших роботах.

2010 Mathematics Subject Classiﬁcation: 37F20, 28A80, 05C63 Keywords: Weierstrass function; non-diﬀerentiability; iterative function systems DOI: http://dx.doi.org/10.15673/tmgc.v12i2.1485

43 44 Cl. David

INTRODUCTION In his seminal paper of 1981, J. E. Hutchinson [8] introduces, for the first time, what will be later qualified of “iterated function system” (I.F.S.), as a finite set of contraction maps, each defined on a compact metric set K of d the euclidean space R , d N‹: P = T1,...,TN ,N N‹ S t u P where N‹ denotes the set of strictly positive integers, such that N K = Ti(K) i=1 ď The compact set K is then said to be “invariant” with respect to the set (one often refer to this result as the “Gluing Lemma”). S A prequel occurence of such maps, under the form of similitudes, can already be found in the Mandelbrot books of 1977 [11], [12]. Hutchinson’s novelty is to consider not the compact K itself, but the set , which arises naturally, in so far as the invariant compact K is fully determinedS by the set , and, interestingly, is also the limit of a sequence of pre-fractal graphs thatS can be built, in an iterative way, thanks to the maps that constitute the set . Following this work, iteratedS function systems were taken up and even more developed by M. F. Barnsley et al. [2], as “a unified way of generat- ing and classifying a broad class of fractals”. As explained by the authors, fractals were “traditionally viewed as being produced by a process of suc- cessive microscopic refinement taken to the limit”, which, of course, makes sense with the geometric representation one may have of fractal sets, since, when looking at smaller and smaller scales, one finds, again and again, the same form. Of course, at stake are specific and classical types of fractals, as Sierpiński gaskets, dragon curves, Cantor sets, Julia curves, etc. For M. F. Barnsley and S. Demko, those fractals are to be seen as the attrac- tors of iterated function systems, which, of course, joins the approach of J. E. Hutchinson. M. F. Barnsley and S. Demko place themselves in a probabilistic approach. Given still a compact metric space K, the related Banach space C(K) of real-valued functions defined on K, with respect to the norm f C(K) f = max f(x) , x K P ÞÑ } }8 t| | P u and a finite collection

w = w1, . . . , wN ,N N‹ t u P of Borel measurable functions from K to K, they deﬁne the set K, w as an iterated function system if and only if there exists an associatedt setu of On fractal properties of Weierstrass-type functions 45 positive real numbers N p1, . . . , pN , i 1,...,N : pi 0, pi = 1 t u @ P t u ą i=1 ÿ such that the operator T on C(K), given, for any f of C(K), by N x K : T (f)(x) = pi (f wi)(x) @ P i=1 ˝ ÿ has the property: T (C(K)) C(K). Ă Treating w as a set-valued function, through

x K : w(x) = w1(x), . . . , w (x) @ P t N u they then naturally introduce, for the i.f.s. K, w , and a given x of K, the related attractor t u n (x) = lim w˝ (x) n + A Ñ 8 in the sense: n lim w˝ (x) (x) = 0. n + 8 Ñ 8 } ´ A } Classical fractal sets as, for instance, the Sierpiński Gasket, ﬁt this deﬁ- nition. In our previous work on the Weierstrass curve [4], which, as exposed, for instance, by A. S. Besicovitch and H. D. Ursell [3], or, a few years later, by B. Mandelbrot [11], bears fractal properties, we showed that the curve could be obtained by means of a sequence a graphs , that (Γ m )m N approximate the studied one. This is done using a familyW of nonlinearP 2 2 C8 maps from R to R , which happen not to be contractions, in the aforementioned classical sense. The nonlinearity does not enable one to resort to the probabilistic approach of M. F. Barnsley and S. Demko, since there does not exist a constant associated set of probabilities. Yet, even if they are not contractions, our maps bear what can be viewed as an equivalent property, since, at each step of the iterative process, they reduce the two-dimensional Lebesgue measures of a given sequence of rectangles covering the curve. This is due to the fact that they correspond, in a sense, to the composition of a contraction of ratio rx in the horizontal direction, and a dilatation of factor ry in the vertical one, with

rx ry 1. ă Such maps are considered in the book of Robert L. Devaney [6], where they play a part in the ﬁrst step of the horseshoe map process introduced by Stephen Smale. 46 Cl. David

The Weierstrass curve is invariant with respect to the set of those maps, which makes it possible to dispose of an equivalent result of the Gluing Lemma. But what deserves to be enlightened, in our case, is that the intrinsic properties of those curious maps make them all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-diﬀerentiabilty of the Weierstrass function, as shown in [5]. All the more is the generaliza- tion to a broader class of applications that could, then, enable one to build everywhere continuous, though nowhere diﬀerentiable, functions, as we will expose it in the sequel.

1. THE CASE OF THE WEIERSTRASS FUNCTION Notation 1.1. In the following, λ and b are two real numbers such that:

0 λ 1, b = Nb N, λ Nb 1. ă ă P ą We deliberatly made the choice to introduce the notation Nb which replaces the initial b, in so far as, to the origins, b is any real number, whereas we deal with the specific case of a natural integer that we consequently choose to denote by Nb, as an echo to the initial b. The Weierstrass function, introduced in 1875 by K. Weierstrass [13], known as one of these so-called pathological mathematical objects, continuous everywhere, while nowhere differentiable, is the sum of the uniformly convergent trigonometric series, defined, for any real number x, by: + 8 n n (x) = λ cos (2πNb x) . W n=0 ÿ Definition 1.2. (Weierstrass Curve). We will call Weierstrass Curve the restriction to [0, 1) R, of the graph of the Weierstrass function, and denote it by Γ . ˆ W Theoretical study. We place ourselves, in the following, in the euclidian plane of dimension 2, referred to a direct orthonormal frame. The usual Cartesian coordinates are (x, y). Property 1.3. (Periodic properties of the Weierstrass function). For any real number x: + + 8 n n n 8 n n (x + 1) = λ cos (2πNb x + 2πNb ) = λ cos (2πNb x) = (x). W n=0 n=0 W ÿ ÿ The study of the Weierstrass function can be restricted to the interval [0, 1). By following the method developed by J. Kigami [10], we approximate the restriction Γ to [0, 1) R, of the Weierstrass Curve, by a sequence of W ˆ On fractal properties of Weierstrass-type functions 47 graphs, built through an iterative process. To this purpose, we introduce 2 2 the iterated function system of the family of C8 maps from R to R :

T0,...,TN 1 t b´ u 2 where, for any integer i belonging to 0,...,Nb 1 and any (x, y) of R : t ´ u x+i x+i Ti(x, y) = , λ y + cos 2π . Nb Nb

N( 1 ( )) b´ Property 1.4. [4]. Γ = Ti(Γ ). W i=0 W Deﬁnition 1.5. (Word on theŤ graph Γ ). Let m be a strictly positive W integer. We will call number-letter any integer i of 0,...,Nb 1 , and word of length = m, on the graph Γ , anyM set of number-letterst ´ u of the form: |M| W = ( ,..., ). M M1 Mm We will write:

T = T 1 T m . M M ˝ ¨ ¨ ¨ ˝ M Deﬁnition 1.6. For any integer i belonging to 0, ..., Nb 1 , let us denote by: t ´ u i 1 2πi Pi = (xi, yi) = N 1 , 1 λ cos N 1 b´ ´ b´ the ﬁxed point of the map Ti. ( ( )) We will denote by V0 the ordered set (according to increasing abscissa), of the points:

P0, ..., PN 1 t b´ u since for any i of 0,...,N 2 : t b ´ u xi xi+1. ď The set of points V0, where, for any i of 0,...,N 2 , the point Pi t b ´ u is linked to the point Pi+1, constitutes an oriented graph (according to increasing abscissa), which we will denote by Γ 0 . In turn, V0 is called the W set of vertices of the graph Γ 0 . For any natural integer m,W we set:

N 1 b´ Vm = Ti (Vm 1) . ´ i=0 ď The set of points Vm, where two consecutive points are linked, is an oriented graph (according to increasing abscissa), which we will denote by

Γ m . Again Vm is called the set of vertices of the graph Γ m . In what W W 48 Cl. David

follows we will denote by mS the number of vertices of the graph Γ m , and write: N W m m m Vm = 0 , 1 ,..., 1 . S S SNmS ´ Property 1.7. For any natural! integer m: )

Vm Vm+1. Ă Property 1.8. For any integer i belonging to 0,...,N 2 : t b ´ u Ti (PN 1) = Ti+1 (P0) . b´

FIGURE 1.1. Fixed points P0, P1, P2, and the graph Γ 0 , 1 W in the case when λ = 2 and Nb = 3.

1 FIGURE 1.2. Graph Γ 1 , in the case when λ = , Nb = 3, W 2 T0(P2) = T1(P0), and T1(P2) = T2(P1).

Deﬁnition 1.9. (Vertices of the graph Γ ). Two points X and Y of Γ will be called vertices of the graph Γ ifW there exists a natural integer mW such that: W (X,Y ) V 2 P m Deﬁnition 1.10. (Consecutive vertices on the graph Γ ). Two points X and Y of Γ will be called consecutive vertices of theW graph Γ if there exist a naturalW integer m, and an integer j of 0,...,N 2 , suchW that: t b ´ u X = (Ti1 ... Tim )(Pj) m ˝ ˝ i1, . . . , im 0,...,Nb 1 # Y = (Ti1 ... Tim )(Pj+1) t u P t ´ u ˝ ˝ On fractal properties of Weierstrass-type functions 49

FIGURE 1.3. Graphs Γ 0 (in green), Γ 1 (in red), Γ 2 (in W W 1 W orange), Γ (in cyan), in the case where λ = and Nb = 3. W 2 or:

X = (Ti1 Ti2 ... Tim )(PN 1),Y = (Ti1+1 Ti2 ... Tim )(P0). ˝ ˝ ˝ b´ ˝ ˝ Property 1.11. The set Vm is dense in Γ . W m N P Deﬁnition 1.12. (Edge relation,Ť on the graph Γ ). Given a natural in- W teger m, two points X and Y of Γ m will be called adjacent if and only if W X and Y are two consecutive vertices of Γ m . We will write: W X Y „m

This edge relation ensures the existence of a word = ( 1,..., m) of length m, such that X and Y both belong to the iterate:M M M

T V0 = (T 1 T m ) V0 M M ˝ ¨ ¨ ¨ ˝ M Given two points X and Y of the graph Γ , we will say that X and Y are adjacent if and only if there exists a naturalW integer m such that: X Y „m Proposition 1.13 (Adresses, on the Weierstrass Curve). Given a strictly positive integer m, and a word = ( 1,..., m) of length m N‹, on M M M P the graph Γ m , for any integer j of 1,...,Nb 2 , any X = T (Pj) of W t ´ u M Vm V0, i.e. distinct from one of the Nb ﬁxed point Pi,(0 i Nb 1), hasz exactly two adjacent vertices, given by: ď ď ´

T (Pj+1) and T (Pj 1) M M ´ where:

T = T 1 T m . M M ˝ ¨ ¨ ¨ ˝ M By convention, the adjacent vertices of T (P0) are T (P1) and T (PN 1), M M M b´ those of T (PN 1), T (PN 2) and T (P0). M b´ M b´ M 50 Cl. David

Notation 1.14. For any integer j belonging to 0,...,Nb 1 , any natural integer m, and any word of length m, we set:t ´ u M

T (Pj) = (x(T (Pj)), y(T (Pj))), M M M 1 Lm = x(T (Pj+1)) x(T (Pj)) = M ´ M (N 1) N m b ´ b hj,m = y(T (Pj+1)) y(T (Pj)). M ´ M

(Pj)

(Pj+1)

Notation 1.15. We will denote by

ln λ D = 2 + W ln Nb the Hausdorﬀ dimension of Γ , see [1], [9]. W Theorem 1.16 (An upper bound and a lower bound, for the box-dimension of the Weierstrass Curve). [4] For any integer j belonging to

0, 1,...,N 2 , t b ´ u each natural integer m, and each word of length m, let us consider the Mm rectangle j,m, m , whose sides are parallel to the horizontal and vertical axes, of width:R M

1 Lm = x(T m (Pj+1)) x(T m (Pj)) = M ´ M (N 1) N m b ´ b and height hj,m , such that the points T m (Pj) and T m (Pj+1) are two vertices of this| rectangle.| We set: M M

(2N 1)λ(N 2 1) 2N η = 2π2 b ´ b ´ + b . W (N 1)2(1 λ)(λN 2 1) (λN 2 1)(λN 3 1) " b ´ ´ b ´ b ´ b ´ * On fractal properties of Weierstrass-type functions 51

2 D 2 sin π min sin π(2 j+1) 2π 1 (Nb 1) ´ W 1 λ N 1 N 1 N (N 1) λN 1 , ´ ´ b´ 0 j Nb 1 b´ ´ b b´ b´ $ " ( ) ď ď ´ ˇ ( )ˇ * ˇ ˇ if is odd, ’ ˇ ˇ Nb ’ ’ C1(Nb) = ’ 2 D 2 π π(2j+1) ’ (Nb 1) ´ W max sin min sin ’ ´ 1 λ Nb 1 0 j N 1 Nb 1 ´ &’ " ´ ´ ď ď b´ ´ ( ) ˇ ˇ 1 N 2 ˇ 2π ˇ 1 4 b´ , 2 ´2 , ’ ˇNb(Nb 1) λNˇ b 1 N N 1 ’ ´ ´ ´ b b ´ ’ * ’ if Nb is even. ’ ’ and: %’ 2 D C2(Nb) = η (Nb 1) ´ W . W ´ Then: 2 D 2 D C1(N ) L ´ W hj,m C2(N ) L ´ W . b m ď | | ď b m Notation 1.17. Given a natural integer m, we set:

(D 2) m 2 D Nb W ´ h = L ´ W = . m m (N 1)2 D b ´ ´ W Then the following inequality holds:

hjm hm. ď

Y ~ X m

X ~ Y m

Corollary 1.18 (of Theorem 1.16). For any natural integer m, any integer j belonging to 0, 1,...,Nb 2 , and each word m+1 of length m + 1, the two-dimensionalt Lebesgue measure´ u M

µ j,m+1, m+1 L R M of the rectangle j,m+1, m+1 , is( such that, for) any integer k belonging to R M 0, 1,...,Nb 2 , any integer ℓ belonging to 0, 1,...,Nb 2 , and each wordt of´ lengthu m: t ´ u Mm

µ ( j,m+1, m+1 ) µ ( ℓ,m, m ). L R M ă L R M 52 Cl. David

Proof. Given a natural integer m, j in 0, 1,...,Nb 2 , and a word of length m + 1, the two-dimensionalt Lebesgue measure´ u of the rec- Mm+1 tangle j,m+1, m+1 can be obtained thanks to the values of the cartesian R M coordinates of the consecutive vertices T m+1 (Pj) and T m+1 (Pj+1): M M

µ ( j,m+1, m+1 ) = x(T m+1 (Pj+1)) x(T m+1 (Pj)) L R M M ´ M ˆ ( ) y(T m+1 (Pj+1)) y(T m+1 (Pj)) . ˆ M ´ M ˇ ˇ One may then write: ˇ ˇ ˇ ˇ

T m+1 = Tk T m , k 0, 1,...,Nb 1 M ˝ M P t ´ u where is a word of length m. Thus, due to: Mm x(T (Pj+1))+k Mm+1 y(T m+1 (Pj+1)) = λ y(T m (Pj+1)) + cos 2π M M Nb ( ( )) x(T (Pj ))+k Mm+1 y(T m+1 (Pj)) = λ y (T m (Pj)) + cos 2π M M Nb ( ( )) and:

x(T m (Pj+1)) x (T m (Pj)) = Lm y(T m (Pj+1)) y(T m (Pj)) M ´ M ď | M ´ M | one has:

y(T m+1 (Pj+1)) y(T m+1 (Pj)) M ´ M ď ˇ λ y(T ˇ m (Pj+1)) y(T m (Pj)) + ˇ ď Mˇ ´ M ˇ 2π ˇ ˇ + x(T m (Pj+1))ˇ x(T m (Pj)) Nb M ´ M ˇ 2π ˇ λ y(T m (Pj+1)) ˇ y(T m (Pj)) + Lm ˇ ď M ´ M Nb ˇ ˇ ˇ 2π ˇ λ + y(T m (Pj+1)) y(T m (Pj)) ď N M ´ M ( b ) ˇ ˇ which yields: ˇ ˇ

Lm µ j,m+1, m+1 = y(T m+1 (Pj+1)) y(T m+1 (Pj)) L R M Nb ˆ M ´ M ( ) ˇ ˇ Lm ˇ ˇ 2π λ y(T m (Pj+1)) y(T m (Pj)) + Lm ď N ˆ M ´ M N b " b * ˇ ˇ Lm ˇ 2π ˇ λ + y(T m (Pj+1)) y(T m (Pj)) . ď N ˆ N M ´ M b ( b ) ˇ ˇ ˇ ˇ On fractal properties of Weierstrass-type functions 53

Due to the symmetric roles played by the integers j and ℓ, one has just to prove the result for j = ℓ. Since:

µ ( j,m, m ) = Lm y(T m (Pj+1)) y(T m (Pj)) L R M ˆ M ´ M and, due to Nb 3, we get that ˇ ˇ ě ˇ ˇ 1 2π 1 2 λ + 1 = 2 λNb +2π Nb 0 Nb Nb ´ Nb ´ ă ( ) " 1 * ă which yield the expected result. loomoon □ 2. A SPECIFIC CLASS OF I.F.S. Weierstrass-type functions have been previously studied, but under the Hausdorff dimension point of view. One may refer, for instance, to the study by B. R. Hunt [7], where the author considers functions defined, for any real number x, by: + 8 WΘ(x) = an g (bn x + θn) n=0 ÿ where an is a positive and convergent series, (bn)n N a positive and in- P creasing sequence, Θ = (θn)n N a uniformly distribed sequence of numbers each belongingř to [0, 1], and playingP the part of arbitrary phases, g being a Lispchitz and 1-periodic function. In the case where the following assumptions are satisfied: (i) there exist two strictly positive real numbers ρ and σ such that:

1 ρ σ, n N : ρ bn bn+1 σ bn ă ă @ P ď ď (ii) there exists D in ]1, 2[ such that: ln a lim n = D 2 n + ln b Ñ 8 n ´ (iii) there exist a positive integer p, a strictly positive real constant M, ℓ a constant ℓ in (0, 1), such that for all δ in σp , ℓ , and for any real number x chosen randomly according to a uniform distribution on [0, 1], the density function of: [ ] x g(x + δ) g(x) p ÞÑ ´ p 1 has a L ´ norm at most equal to M. B. R. Hunt [7] shows that for almost every Θ in [0, 1]8, the graph of WΘ has Hausdorff dimension D. It happens that in the case of such functions, the Hausdorff dimension is equal to the box-dimension. Yet, as concerns the lower bound estimate required to obtain the explicit value of the Hausdorff/box dimension, the author calls for strictly positive 54 Cl. David constants K and K1 which, as in existing earlier works, are not given ex- plicitely (see, in the Hunt study, [7, section 3., page 798]). Moreover, no relation is made with the non-differentiability of such functions. One may also note that such functions cannot be described by means of a finite iterated function systems, which does not allow any use of the Gluing Lemma. In addition, the fact that the author considers, very generally, Lispchitz functions g is not specifically justified. It is all the more interesting as evoked in the above since, if the functions g were contractant ones, one falls back more easily on classical configurations. In fact, one may just consider the limit case of functions satisfying a Lipschitz condition with a Lipschitz constant of value 1. What seemed of interest to us was to generalize our results to, indeed, a class of Weierstrass-type functions, but defined through an iterated function system which would bear analogous properties of the maps Ti, 0 i Nb 1. First, the box-dimension can be obtained rather simply, withoutď ď calling´ for theoretical background in dynamic systems theory, just by applying a similar method as in [4]. Then, one can also simply prove the non- differentiability of such functions, as in [5].

Notation 2.1. In the sequel: (i) N is a strictly positive integer, greater than 2; (ii) T and M are strictly positive real numbers; N N (iii) (αi)0 i N 1 0, ,N 1 and (βi)0 i N 1 0,...,N 1 are orderedď ď ´ setsP t of¨ positive ¨ ¨ ´ integers:u ď ď ´ P t ´ u

i 0,...,N 2 : αi αi+1, βi βi+1 @ P t ´ u ď ď (iv) ψ is a T -periodic, bounded function from R to R satisfying a Lipschitz condition; (v) ry is a real number such that:

0 ry 1, ryN 1. ă ă ą (vi) We set: 1 r = . x N

(vii) ϕ0, . . . , ϕN 1 and φ0, . . . , φN 1 are sets of aﬃne contractive maps t ´ u t ´ u from R to R, of respective ratios rx and ry, deﬁned, for any integer i of 0,...,N 1 , and for any real number x: t ´ u

ϕi(x) = rx(x + αi), φi(x) = ry(x + βi). On fractal properties of Weierstrass-type functions 55

(viii) We denote by ψ0, . . . , ψN 1 the set of maps from R to R such that, for any integert i of 0,...,N´ u 1 : t ´ u ψi = ψ ϕi. ˝ Notation 2.2. We introduce the set of maps from R2 to R2

T0,..., TN 1 t ´ u such that, for any integer i of 0,...,N 1 , and any (x, y) of R2: tr r ´ u Ti(x, y) = ϕi(x), φi(y) + ψi(x) . Definition 2.3. We introduce( the -type function), defined, for any real r number x, by: W + 8 n n (x) = ry ψ(TN x). W n=0 ÿ Property 2.4. For anyĂ real number x, the series: + 8 n n (x) = ry ψ(TN x). W n=0 ÿ is convergent Ă Proof. One may simply note that for any real number x: n n n ry ψ(TN x) ry sup ψ(t) À t R | | P ˇ ˇ + ˇ ˇ 8 n which yields the expected result, since ry is a geometric convergent n=0 series. ř □ Definition 2.5. We will call -type curve the restriction to [0,T ) R, of the graph of the -type function,W and denote it by Γ . ˆ W W 2.6. Theoretical study. We place ourselves, in the following, in the eu- Ă clidian plane of dimension 2, referred to a direct orthonormal frame. The usual Cartesian coordinates are (x, y).

Property 2.7. For any integer i of 0,...,N 1 , the map Ti admits a t ´ u ﬁxed point, that we will denote by Pi: r αi βi 1 αi Pi = , r+ ψi . N 1 1 ry 1 ry N 1 ( ´ ´ ´ ( ´ )) r Lemma 2.8. For any integer i belonging to 0,...,N 1 , the map Ti is t ´ u a bijection of the Weierstrass-type curve on R. 56 Cl. David

Proof. Let us consider i 0,...,N , a point (y, (y)) of Γ , and let us look for a real number x suchP t that: u W W Ă T x, (x) = y, (y) . i W W One has: ( ) ( ) Ă Ă y = ϕi(x) = rx (x + αi) which yields: 1 x = r´ y αi. x ´ This enables one to obtain: + 1 8 n n+1 n (x) = (rx´ y αi) = ry ψ TN y T αi N i = W W ´ n=0 ´ ÿ ( ) Ă + 8 n n+1 = ry ψ TN y n=0 ÿ ( ) due to the T -periodicity of the function ψ, which leads to: n+1 n n+1 ψ TN y T αi N i = ψ TN y ´ since αi, N and i are( integers. Also: ) ( ) 1 1 Ti x, (x) = ϕi(r´ y αi), φi( (x)) + ψi(r´ y αi) W x ´ W x ´ ( ) ( + ) r Ă 8 n n+1Ă = y, ry ry ψ(TN y) + ψ(T y) ( n=0 ) + ÿ 8 n+1 n+1 = y, ry ψ(TN y) + ψ(T y) ( n=0 ) +ÿ 8 n n = y, ry ψ(TN y) ( n=0 ) ÿ = y, (y) . W There exists thus a unique( real) number x such that: Ă T x, (x) = y, (y) . i W W Lemms is completed. ( ) ( ) □ Ă Ă Theorem 2.9 (An upper bound and a lower bound for the box-dimension of the Weierstrass-type Curve). For any j 0, 1,...,N 2 , each natural integer m, and each word of length mP, t let us consider´ u the rectangle, whose sides are parallel to theM horizontal and vertical axes, of width: m Lm = x(T (Pj+1)) x(T (Pj)) = rx M ´ M On fractal properties of Weierstrass-type functions 57 and height hj,m , such that the points T (Pj) and T (Pj+1) are two vertices of this| rectangle.| We set: M M

C (N) = 1 min (β β ) + ψ αi+1 ψ αi 1 1 ry i+1 i i+1 N 1 i N 1 ´ 0 j N 1 ´ ´ ´ ´ ´ ď ď ´ ! ) ␣ ( r)x 1( αj)+1( αj rx ´ , ´ ry 1 N 1 ´ ry ´ and:

βj+1 βj 1 αj+1 αj αj+1 αj C2(N) = | ´ | + + | ´ | . 1 r 1 r N 1 ´ N 1 N (r r ) y y ˇ ˇ y x ´ ´ ˇ ´ ´ ˇ ´ If: ˇ ˇ ˇ ˇ C1(N) 0 ě one has: 2 D 2 D C1(N) Lm´ W hj,m C2(N) Lm´ W . ď | | ď where: Ă Ă ln r D = 2 + y . W ln N which yields the fractal characterĂ of the Weierstrass-type Curve, the box- dimension of which is then D . W Proof. The proof is obtainedĂ as in [4]. It is based on the fact that, given a strictly positive integer m, and two points X and Y of Vm such that: X Y „m there exists a word of length = m, on the graph Γ , and an integer M |M| j of 0,...,N 2 2, such that: W t ´ u Ă X = T (Pj), Y = T (Pj+1). M M By writing T under the form: M r r

T = Tim Tim 1 ... Ti1 r M ˝ ´ ˝ ˝ m where (i1, . . . , im) 0,...,N 1 , one gets: P t r r´ u r r m m N k N k x T (Pj) = rx xj + rx αk, x T (Pj+1) = rx xj+1 + rx αk M M k=1 k=1 ( ) ÿ ( ) ÿ and r r m k m m k k ℓ y T (Pj) = ry yj + ry ´ ψk rx xj + rx αm ℓ , M ´ k=1 ( ℓ=0 ) ( ) ÿ ÿ r 58 Cl. David

m k m m k k ℓ y T (Pj+1) = ry yj+1 + ry ´ ψk rx xj+1 + rx αm ℓ M ´ k=1 ( ℓ=0 ) ( ) ÿ ÿ This leadsr to m k m m k k ℓ hj,m ry (yj+1 yj) = ry ´ ψik rx xj+1 + rx αm ℓ ´ ´ ´ ´ k=1 ℓ=0 ÿ " ( ÿ ) k k ℓ ψik rx xj + rx αm ℓ , ´ ´ ℓ=0 ( ÿ )* where rm m m βj+1 βj y αj+1 αj ry (yj+1 yj) = ry ´ + ψj+1 N 1 ψj N 1 . ´ 1 ry 1 ry ´ ´ ´ ´ ´ ! ( ) ( )) Since the maps ψik , 1 k m, satisfy a Lipschitz condition, with a Lipschitz constant equal toď1, oneď has thus: m y(T (Pj)) y(T (Pj+1)) ry (yj+1 yj) M ´ M ´ ´ ď m m ˇ m k k ˇ m k k αj+1 αj ˇ r ´ r xj+1 xj ˇ= r ´ r | ´ | = ď y x | ´ | y x N 1 k=1 k=1 ÿ ÿ ´ m rx 1 m m rx ´ ry αj+1 αj m rx 1 αj+1 αj = ry rx | ´ | ry rx | ´ |, ry 1 N 1 ď ry 1 N 1 ´ ry ´ ´ ry ´ which leads to

m βj+1 βj y(T (Pj)) y(T (Pj+1)) ry ´ + M ´ M ě 1 ry ´ rm rm rm y αi+1 αi y x αj+1 αj + ψi+1 N 1 ψi N 1 rx ´ | ´ |. 1 ry ´ ´ ´ ´ ry rx N 1 ´ ! ( ) ( )) ´ ´ If

1 αj+1 αj min (βj+1 βj) + ψj+1 N 1 ψj N 1 1 ry 0 j N 1 ´ ´ ´ ´ ´ ´ ď ď ´ ! ! ( ) ( ))) rx 1 αj+1 αj rx | ´ | 0 ´ ry 1 N 1 ě ´ ry ´ due to the symmetric roles played by T (Pj) and T (Pj+1), one may only consider the case when M M m βj+1 βj y(T (P )) y(T (P )) r ´ + j j+1 y 1 ry M ´ M ě ´ m ry αi+1 α m r 1 αj+1 αj + ψ ψ i r x | ´ | 0. 1 ry i+1 N 1 i N 1 y ry rx N 1 ´ ´ ´ ´ ´ 1 r ´ ě ! ( ) ( )) ´ y On fractal properties of Weierstrass-type functions 59 which yields y(T (Pj)) y(T (Pj+1)) M ´ M ě

m 1 αi+1 αi ry min (βi+1 βi) + ψi+1 N 1 ψi N 1 ě 1 ry 0 j N 1 ´ ´ ´ ´ ´ " ´ ď ď ´ ! ! ( ) ( ))) rx 1 αj+1 αj r ´ . ´ ry 1 x N 1 ´ ry ´ * The predominant term is thus

m m (D 2) ln N m (D 2) 2 D 2 D r = e ´ = N ´ = Lm´ W (N 1) ´ y W W ´ W Ă Ă Ă Ă One also has

m hj,m r yj+1 yj + | | ď y | ´ | m k k m k k ℓ k ℓ + ry ´ ψik rx xj+1 + rx αm ℓ ψik rx xj + rx αm ℓ ´ ´ ´ k=1 ˇ ℓ=0 ℓ=0 ˇ ÿ ˇ ( ÿ ) ( ÿ )ˇ ˇ rm ˇ m βj+1 ˇ βj y αj+1 αj ˇ ry | ´ | + ψj+1 N 1 ψj N 1 ď 1 ry 1 ry ´ ´ ´ ´ ´ ˇ ( ) ( )ˇ m ˇk ˇ k m k k ˇ ℓ k ˇ ℓ + ry ´ ψik rx xj+1 + rx αm ℓ ψik rx xj + rx αm ℓ ´ ´ ´ k=1 ˇ ℓ=0 ℓ=0 ˇ ÿ ˇ ( ÿ ) ( ÿ )ˇ ˇ rm m ˇ m βj+1 ˇβj y αj+1 αj m k k ˇ r | ´ | + + r ´ r xj+1 xj ď y 1 ry 1 r N 1 ´ N 1 y x | ´ | ´ y ´ ´ k=1 ´ ˇ ˇ ÿ ˇ ˇ rm m ˇ ˇ x r 1 m m βj+1 βj y αj+1 αj m rx ry = r | ´ | + + x x r ´ y 1 ry N 1 N 1 j+1 j y rx ´ 1 ry ´ ´ ´ | ´ | ry 1 ´ ˇ ˇ ´ ry ˇ ˇ rm rm rm m βj+1 βj y ˇ αj+1 αj ˇ y x = r | ´ | + + x x r ´ y 1 ry N 1 N 1 j+1 j x ´ 1 ry ´ ´ ´ | ´ | ry rx ´ ˇ ˇ ´ rm ˇ ˇ rm m βj+1 βj y ˇ αj+1 αj ˇ y r | ´ | + + x x r y 1 ry N 1 N 1 j+1 j x ď ´ 1 ry ´ ´ ´ | ´ | ry rx ´ ˇ ˇ ´ rm ˇ ˇ m βj+1 βj y ˇ αj+1 αj ˇ m αj+1 αj r | ´ | + + r | ´ | . y 1 ry N 1 N 1 y ď ´ 1 ry ´ ´ ´ N (ry rx) ´ ˇ ˇ ´ ˇ ˇ Since ˇ ˇ ln ry (D 2) ln N (D 2) D = 2 + , ry = e W ´ = N W ´ , W ln N Ă Ă Ă 60 Cl. David one has thus:

m βj+1 βj 1 αj+1 αj αj+1 αj hj,m r | ´ | + + | ´ | . | | ď y 1 r 1 r N 1 ´ N 1 N (r r ) " y y ˇ ˇ y x * ´ ´ ˇ ´ ´ ˇ ´ Theorem 2.9 is proved. ˇ ˇ □ ˇ ˇ Corollary 2.10. The -type functions are non-diﬀerentiable. W Proof. One has simply to use the analogous density property as in 1.11. Given a natural integer m, and two points X = (x, (x)), Y = (y, (y)) of the pre-fractal graph Γ Γ such that: W W m Ă W W Ă Ă x y, X Y Ă ď Ă „m one may write:

X = T (Pk),Y = (x + Lm, (x + Lm)) = T (Pk+1) Mm,j W Mm,j m where m,j, 0 j N 1 is a word of length m, while k denotes an integerM of ther setď 0,...,Nď ´ 2 . r One may note thatt ´ u 1 x(T (Pk)) x(T (Pk+1)) = = Lm 0. Mm,j ´ Mm,j N m mÝÑ+ ˇ ˇ Ñ 8 Thus ˇ ˇ ˇ r r ˇ 2 D x(T (Pk)) x(T (Pk+1)) C1(N) Lm´ W = Mm,j Mm,j W ´ W ě Ă ˇ ˇ 2 D ˇ ( ) ( )ˇ ´ W ˇĂ r Ă= Cr1(N) x(T m,jˇ(Pk)) x(T m,j (Pk+1)) , M ´ M Ă ˇ ˇ which leads to ˇ ˇ ˇ r r ˇ x(T (Pk)) x(T (Pk+1)) W Mm,j ´W Mm,j x T (Pk) x T (Pk+1) ě ˇ Ă( r Mm,j )´ Ă(Mm,jr )ˇ ˇ ˇ 1 D ˇ ( r ) ( r ) ˇ ´ W ˇ C1(N) x(T ˇ m,j (Pk)) x(T m,j (Pk+1)) = ě M ´ M Ă ˇ ˇ 1 D ˇ = C (Nˇ ) L ´ W , ˇ r r 1 ˇ m Ă where ln r ln(r N) 1 D = 1 y = y 0 ´ W ´ ´ ln N ´ ln N ă By passing to the limitĂ when the integer m tends towards inﬁnity, one gets the non-diﬀerentiability expected result:

(x + Lm) (x) lim W ´ W = + . m + ˇ Lm ˇ 8 Ñ 8 ˇ ˇ ˇ Ă Ă ˇ ˇ ˇ ˇ ˇ On fractal properties of Weierstrass-type functions 61 where: lim Lm = 0. m + Ñ 8 Corollary is completed. □ REFERENCES [1] Krzysztof Barański, Balázs Bárány, Julia Romanowska. On the dimension of the graph of the classical Weierstrass function. Adv. Math., 265:32–59, 2014, doi: 10.1016/j.aim.2014.07.033. [2] M. F. Barnsley, S. Demko. Iterated function systems and the global con- struction of fractals. Proc. Roy. Soc. London Ser. A, 399(1817):243–275, 1985, doi: 10.1098/rspa.1985.0057. [3] A. S. Besicovitch, H. D. Ursell. Sets of fractional dimensions (V): on dimensional numbers of some continuous curves. J. London Math. Soc., s1-12(1):18–25, 1937, doi: 10.1112/jlms/s1-12.45.18. [4] Claire David. Bypassing dynamical systems: a simple way to get the box-counting dimension of the graph of the Weierstrass function. Proc. Int. Geom. Cent., 11(2):53– 68, 2018, doi: 10.15673/tmgc.v11i2.1028. [5] Claire David. Wandering across the Weierstrass function, while revisiting its properties. To appear, 2019. [6] Robert L. Devaney. An introduction to chaotic dynamical systems. Studies in Nonlin- earity. Westview Press, Boulder, CO, 2003. Reprint of the second (1989) edition. [7] Brian R. Hunt. The Hausdorﬀ dimension of graphs of Weierstrass functions. Proc. Amer. Math. Soc., 126(3):791–800, 1998, doi: 10.1090/S0002-9939-98-04387-1. [8] John E. Hutchinson. Fractals and self-similarity. Indiana Univ. Math. J., 30(5):713– 747, 1981, doi: 10.1512/iumj.1981.30.30055. [9] Gerhard Keller. A simpler proof for the dimension of the graph of the classical Weierstrass function. Ann. Inst. Henri Poincaré Probab. Stat., 53(1):169–181, 2017, doi: 10.1214/15-AIHP711. [10] Jun Kigami. A harmonic calculus on the Sierpiński spaces. Japan J. Appl. Math., 6(2):259–290, 1989, doi: 10.1007/BF03167882. [11] Benoit B. Mandelbrot. Fractals: form, chance, and dimension. W. H. Freeman and Co., San Francisco, Calif., revised edition, 1977. Translated from the French. [12] Benoit B. Mandelbrot. The fractal geometry of nature. W. H. Freeman and Co., San Francisco, Calif., 1982. Schriftenreihe für den Referenten. [13] K. Weierstrass. Über kontinuierliche Funktionen eines reellen arguments, die für keinen Wert des letzteren einen bestimmten Diﬀerentialquotienten besitzen. Journal für die reine und angewandte Mathematik, 79:29–31, 1875.

Claire David SORBONNE UNIVERSITÉ, CNRS, LABORATOIRE JACQUES-LOUIS LIONS, 4, PLACE JUSSIEU 75005, PARIS, FRANCE Email: [email protected] ORCID: orcid.org/0000-0002-4729-0733