On fractal properties of Weierstrass-type functions

. In the sequel, starting from the classical Weierstrass function deﬁned, for any real number x , by W ( x ) = nb x ) , where λ and N b are two real numbers such that 0 ă λ ă 1 , N b P N and λ N b ą 1 , we highlight intrinsic properties of curious maps which happen to constitute 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.


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 the euclidean space R d , d P N ‹ : S = tT 1 , . . . , T N u , N P N ‹ where N ‹ denotes the set of strictly positive integers, such that positive real numbers tp 1 , . . . , p N u , @ i P t1, . . . , N u : p i ą 0, such that the operator T on C(K), given, for any f of C(K), by has the property: T (C(K)) Ă C(K). Treating w as a set-valued function, through @ x P K : w(x) = tw 1 (x), . . . , w N (x)u they then naturally introduce, for the i.f.s. tK, wu, and a given x of K, the related attractor in the sense: lim nÑ+8 }w˝n(x)´A(x)} 8 = 0.
Classical fractal sets as, for instance, the Sierpiński Gasket, fit this definition.
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 (Γ W m ) mPN , that approximate the studied one. This is done using a family of nonlinear C 8 maps from R 2 to R 2 , 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 r x in the horizontal direction, and a dilatation of factor r y in the vertical one, with r x r y ă 1.
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. 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 generalization 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.

THE CASE OF THE WEIERSTRASS FUNCTION
Notation 1.1. In the following, λ and b are two real numbers such that: We deliberatly made the choice to introduce the notation N b 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 N b , 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: 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 W to [0, 1)ˆR, of the Weierstrass Curve, by a sequence of graphs, built through an iterative process. To this purpose, we introduce the iterated function system of the family of C 8 maps from R 2 to R 2 : tT 0 , . . . , T N b´1 u where, for any integer i belonging to t0, . . . , N b´1 u and any (x, y) of R 2 : ) . We will write: Definition 1.6. For any integer i belonging to t0, ..., N b´1 u, let us denote by: the fixed point of the map T i . We will denote by V 0 the ordered set (according to increasing abscissa), of the points: tP 0 , ..., P N b´1 u since for any i of t0, . . . , N b´2 u: The set of points V 0 , where, for any i of t0, . . . , N b´2 u, the point P i is linked to the point P i+1 , constitutes an oriented graph (according to increasing abscissa), which we will denote by Γ W 0 . In turn, V 0 is called the set of vertices of the graph Γ W 0 .
For any natural integer m, we set: The set of points V m , where two consecutive points are linked, is an oriented graph (according to increasing abscissa), which we will denote by Γ W m . Again V m is called the set of vertices of the graph Γ W m . In what 48 Cl. David follows we will denote by N S m the number of vertices of the graph Γ W m , and write: ) .

Property 1.7.
For any natural integer m: Property 1.8. For any integer i belonging to t0, . . . , N b´2 u: Definition 1.9. (Vertices of the graph Γ W ). Two points X and Y of Γ W will be called vertices of the graph Γ W if there exists a natural integer m such that: (X, Y ) P V 2 m Definition 1.10. (Consecutive vertices on the graph Γ W ). Two points X and Y of Γ W will be called consecutive vertices of the graph Γ W if there exist a natural integer m, and an integer j of t0, . . . , N b´2 u, such that: or: Given two points X and Y of the graph Γ W , we will say that X and Y are adjacent if and only if there exists a natural integer m such that: has exactly two adjacent vertices, given by: where: By convention, the adjacent vertices of T M (P 0 ) are T M (P 1 ) and T M (P N b´1 ), those of T M (P N b´1 ), T M (P N b´2 ) and T M (P 0 ). Notation 1.14. For any integer j belonging to t0, . . . , N b´1 u, any natural integer m, and any word M of length m, we set: Notation 1.15. We will denote by the Hausdorff dimension of Γ W , see [1], [9].
and height |h j,m |, such that the points T M m (P j ) and T M m (P j+1 ) are two vertices of this rectangle. We set: On fractal properties of Weierstrass-type functions 51 and: Then: Notation 1.17. Given a natural integer m, we set: Then the following inequality holds: of the rectangle R j,m+1,M m+1 , is such that, for any integer k belonging to t0, 1, . . . , N b´2 u, any integer ℓ belonging to t0, 1, . . . , N b´2 u, and each word M m of length m: Proof. Given a natural integer m, j in t0, 1, . . . , N b´2 u, and a word M m+1 of length m + 1, the two-dimensional Lebesgue measure of the rectangle R j,m+1,M m+1 can be obtained thanks to the values of the cartesian coordinates of the consecutive vertices T M m+1 (P j ) and T M m+1 (P j+1 ): One may then write: where M m is a word of length m. Thus, due to: and: x(T M m (P j+1 ))´x (T M m (P j )) = L m ď |y(T M m (P j+1 ))´y(T M m (P j ))| one has:ˇy (T M m+1 (P j+1 ))´y(T M m+1 (P j ))ˇˇď ď λˇˇy(T M m (P j+1 ))´y(T M m (P j ))ˇˇ+ hich yields: )ˇˇy (T M m (P j+1 ))´y(T M m (P j ))ˇˇ. Due to the symmetric roles played by the integers j and ℓ, one has just to prove the result for j = ℓ. Since: which yield the expected result. □

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: where ř a n is a positive and convergent series, (b n ) nPN a positive and increasing sequence, Θ = (θ n ) nPN a uniformly distribed sequence of numbers each belonging to [0, 1], and playing 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: , and for any real number x chosen randomly according to a uniform distribution on [0, 1], the density function of: has a L p p´1 norm at most equal to M . B. R. Hunt [7] shows that for almost every Θ in [0, 1] 8 , the graph of W Θ 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 constants K and K 1 which, as in existing earlier works, are not given explicitely (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 T i , 0 ď i ď N b1 . First, the box-dimension can be obtained rather simply, without calling for theoretical background in dynamic systems theory, just by applying a similar method as in [4]. Then, one can also simply prove the nondifferentiability 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; (iii) (α i ) 0ďiďN´1 P t0,¨¨¨, N´1u N and (β i ) 0ďiďN´1 P t0, . . . , N´1u N are ordered sets of positive integers: (iv) ψ is a T -periodic, bounded function from R to R satisfying a Lipschitz condition; (v) r y is a real number such that: 0 ă r y ă 1, r y N ą 1.
(vi) We set: (vii) tϕ 0 , . . . , ϕ N´1 u and tφ 0 , . . . , φ N´1 u are sets of affine contractive maps from R to R, of respective ratios r x and r y , defined, for any integer i of t0, . . . , N´1u, and for any real number x: On fractal properties of Weierstrass-type functions 55 (viii) We denote by tψ 0 , . . . , ψ N´1 u the set of maps from R to R such that, for any integer i of t0, . . . , N´1u:

Notation 2.2.
We introduce the set of maps from R 2 to R 2 t r T 0 , . . . , r T N´1 u such that, for any integer i of t0, . . . , N´1u, and any (x, y) of R 2 : ) .

Definition 2.3.
We introduce the W-type function, defined, for any real number x, by: r n y ψ(T N n x).

Property 2.4.
For any real number x, the series: is convergent Proof. One may simply note that for any real number x:ˇr n y ψ(T N n x)ˇˇÀ r n y sup tPR |ψ(t)| which yields the expected result, since +8 ř n=0 r n y is a geometric convergent series. □ Definition 2.5. We will call W-type curve the restriction to [0, T )ˆR, of the graph of the W-type function, and denote it by Γ Ă W . 2.6. 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 2.7. For any integer i of t0, . . . , N´1u, the map r T i admits a fixed point, that we will denote by r P i : ) .

Lemma 2.8.
For any integer i belonging to t0, . . . , N´1u, the map T i is a bijection of the Weierstrass-type curve on R.
Proof. Let us consider i P t0, . . . , N u, a point (y, W(y)) of Γ Ă W , and let us look for a real number x such that: ) .
One has: y = ϕ i (x) = r x (x + α i ) which yields: x = r´1 x y´α i . This enables one to obtain: due to the T -periodicity of the function ψ, which leads to: since α i , N and i are integers. Also: ) .
There exists thus a unique real number x such that: ) .
Lemms is completed. and height |h j,m |, such that the points T M (P j ) and T M (P j+1 ) are two vertices of this rectangle. We set: , and: If: one has: where: which yields the fractal character of the Weierstrass-type Curve, the boxdimension of which is then D Ă W . Proof. The proof is obtained as in [4]. It is based on the fact that, given a strictly positive integer m, and two points X and Y of V m such that: there exists a word M of length |M| = m, on the graph Γ Ă W , and an integer j of t0, . . . , N´2u 2 , such that: By writing r T M under the form: where (i 1 , . . . , i m ) P t0, . . . , N´1u m , one gets: This leads to ) .
Since the maps ψ i k , 1 ď k ď m, satisfy a Lipschitz condition, with a Lipschitz constant equal to 1, one has thus:ˇy (T M (P j ))´y(T M (P j+1 ))´r m y (y j+1´yj )ˇˇď ě 0 due to the symmetric roles played by T M (P j ) and T M (P j+1 ), one may only consider the case when y(T M (P j ))´y(T M (P j+1 )) ě r m y β j+1´βj On fractal properties of Weierstrass-type functions 59 which yields The predominant term is thus One also has |h j,m | ď r m y |y j+1´yj |+ Proof. One has simply to use the analogous density property as in 1.11. Given a natural integer m, and two points X = (x, Ă W(x)), Y = (y, Ă W(y)) of the pre-fractal graph Γ Ă W m Ă Γ Ă W such that: x ď y, X " m Y one may write: where M m,j , 0 ď j ď N m´1 is a word of length m, while k denotes an integer of the set t0, . . . , N´2u.
ThušˇˇĂ W ( x( r T M m,j (P k )) )´Ă W ( x( r T M m,j (P k+1 )) )ˇˇˇě )ˇˇˇě ě C 1 (N )ˇˇx( r T M m,j (P k ))´x( r T M m,j (P k+1 ))ˇˇ1´D n r y ln N =´l n(r y N ) ln N ă 0 By passing to the limit when the integer m tends towards infinity, one gets the non-differentiability expected result: Corollary is completed. □