Analysis of a Fractal Boundary: The Graph of the Knopp Function
A usual classification tool to study a fractal interface is the computation of its fractal dimension. But a recent method developed by Y. Heurteaux and S. Jaffard proposes to compute either weak and strong accessibility exponents or local regularity exponents (the so-called p-exponent). These exponents describe locally the behavior of the interface. We apply this method to the graph of the Knopp function which is defined for as , where and . The Knopp function itself has everywhere the same p-exponent . Nevertheless, using the characterization of the maxima and minima done by B. Dubuc and S. Dubuc, we will compute the p-exponent of the characteristic function of the domain under the graph of F at each point and show that p-exponents, weak and strong accessibility exponents, change from point to point. Furthermore we will derive a characterization of the local extrema of the function according to the values of these exponents.
At the beginning of the century several examples of nondifferentiable functions were studied, such as the Weiertrass function or the example we will focus on in the following, that is, the Takagi-Knopp or so called Knopp function (see  and references therein for a review). The issue was the study of the regularity.
Indeed, in 1918, Knopp  introduced a new family of nondifferentiable functions defined on the interval . Going beyond the construction of Weierstrass of a continuous nondifferentiable function, his goal was to build examples of continuous functions for which one-sided limits of the difference quotient at all points do not exist. He considered the function given by the series, for , where , , and is an integer such that .
For and , this function can be seen as a series expanded in the Faber-Schauder basis , , , where is the Schauder function defined by if and 0 elsewhere. In fact We will write for in the following.
Thus, for example, using the characterization of Lipschitz spaces with the help of coefficients in the Schauder-basis , one gets immediately the fact that belongs to .
A further step to study the regularity of this function can be to follow the ideas developed in multifractal analysis. The goal in multifractal analysis is to study the sets of points where the function has a given pointwise regularity, and doing so to check if the regularity changes from point to point and quantify these changes. Recall the definition of Hölder pointwise regularity and local regularity.
Definition 1. Let and . A locally bounded function belongs to if there exists and a polynomial with degree (integer part of ), such that on a neighborhood of , The pointwise Hölder exponent of at is .
Definition 2 (see ). Let . Let and such that . Let be a function in . The function belongs to if there exists , a polynomial with , and such that
The -exponent of at is .
Then again with the help of the Faber-Schauder basis one can prove that, for all , is in (details for this technique can be found in ). It is then easy to check that actually at all . Thus, from the point of view of various notions of regularity, even if it is not differentiable, the function is rather “regular” since one can compute at each point the same regularity exponent. This remark was actually the starting point of this work.
Indeed obviously the graph of the function has a very irregular behavior, and it has also some self-similarity properties. What can we say on the domain under the graph of ?
Denote in the following by the characteristic function of , which takes the value 1 on and 0 outside .
A first reflex is to compute fractal dimensions of the boundary . The box dimension of the graph can be derived by standard methods (see Tricot ) and is exactly . Let us mention that Ciesielski [7, 8] proved results of this type for Schauder and Haar bases expansions in the case of more general families of functions. Jaffard  and Kamont and Wolnik  obtained then general formulas that allow deriving the box dimensions of the graphs of arbitrary functions from their wavelet expansions.
For what concerns the Hausdorff dimension of the graph of , as far as we know, the question is not solved yet in its all generality. It was proved by Ledrappier  in 1992 to be in the special case where is an Erdös number. By the results of Solomyak  on Erdös numbers this amounts to have the computation for almost every in .
Beside the computation of the box and Hausdorff dimension, which provide global quantities to describe the graph of the function, several methods were recently developed to classify fractal boundaries with the help of pointwise exponents. The idea was to be able to give a finer description of the geometry of the boundary, since the pointwise behavior was studied. In , Jaffard and Mélot focused on the computation of the dimension of the set of points where has a given -exponent in the sense of Definition 2. In , Heurteaux and Jaffard studied pointwise exponents more related to the geometry. These are the exponents we are actually interested in.
Indeed denote by the Lebesgue measure in and the dimensional open ball of center and radius . Heurteaux and Jaffard  gave the following definitions.
Definition 3. Let be a domain of and let . The point is weak -accessible in if there exist and such that The supremum of all the values of such that (5) holds is called the weak accessibility exponent in at . We denote it by .
Example 4. Let and . Denote by the complement of . Then one can easily check that at each point of the boundary we have and at we have and .
Definition 5. Let be a domain of and let . The point is strong -accessible in if there exist and such that The infimum of all the values of such that (6) holds is called the strong accessibility exponent in at . We denote it by .
The following proposition is given in .
Proposition 6. Let . Then
Obviously . We will see that thanks to our result one can prove that these two exponents can be different.
Tricot  proved that these exponents are related to local dimension computation. Let us mention, without giving too much details, the relationship of this work  with these exponents. Indeed the author focuses on the formula with some “set function” and the set of closed balls centered on .
Given an open set such that the special choice of leads to definitions of Hausdorff, exterior and interior dimensions, and Packing, exterior and interior dimensions.
The following characterization, written for the setting we are interested in, holds.
Theorem 7 (see ). Let be a bounded open set in with boundary such that . Let . Let and
Then with the Hausdorff interior dimension and the Packing Hausdorff dimension.
We clearly have and . Let us stress that in the setting of Tricot and with the complementary of in . We rather refer to  for more details on local dimensions in their all generality.
We will compute these quantities at the points of the boundary of , where is the function defined by (2). For that we will use the characterization of the maxima and minima done in . This will yield the -exponent at each point of . We will actually derive the fact that the set of local extrema of the function is fully characterized by the set of points where this -exponent has a given value.
We will also prove that the weak and strong accessibility exponents in and change from point to point on the graph of . They also help to provide exact characterization of the sets of local maxima and local minima. Finally we will prove that there is a set of nontrivial Hausdorff dimensions such that the strong accessibility exponents in and are the same and the weak and strong accessibility exponents are different.
Let us emphasize that this is to our knowledge the first time that the computation of these exponents was done in a nearly exhaustive study on a given example. The characterization we get for the set of extrema raise several questions: is it a general property? Do other functions share it? Could it lead to a finer classification of functions in Hölder classes? We would like to address them in future works.
Let us come back now to our work. The outline of the paper is the following. In Section 2 we set our main result. In Section 3 some notations, preliminary remarks and technical lemmas, help us to prepare Section 4 where are the main proofs.
2. Main Results
2.1. Statement of Our Main Result
Our goal is to prove the following Theorem.
Theorem 8 (the main theorem). Let with and . Let be the function defined by (2).
Let and let .
Then, at each point of , the graph of , one has(1) if and only if is a local extremum of . Furthermore(a) if and only if is a local maximum of . And in this case .(b) if and only if is a local minimum of . And in this case .(2)In the other cases where is not a local extremum of , one has .(3)Furthermore one can find a subset such that and for all . The orthogonal projection of on has the Hausdorff dimension .
3. Useful Notations and Results
3.1. Lemmas for Practical Computation of the Exponents
From the computation of the weak accessibility exponent in and it is easy to derive the -exponent. In , Jaffard and Mélot proved that if and only if either is weak -accessible in or is weak -accessible in . As a consequence we have
We will also need the following lemma.
Lemma 9. Let be in with and the domain below (resp., above) the graph of . Consider . Then is strong , accessible in both and .
Proof. Suppose that is the domain below the graph of . Without any loss of generality, we can assume that . Let . Since is in and then there exists a constant such that in neighborhood of
Obviously (resp., ) is greater than the area above (resp., below) the graph of and below (resp., above) the square of side and center .
The same results hold if is the domain above the graph of (we have just to replace by ).
One of our goals for the points which are not extrema of will be to find sequences of local maxima or minima such that the following key lemma holds.
Lemma 10. Let be in and the domain below the graph of . Consider . Suppose that there exists , a sequence of positive numbers, such that as , , and , such that Then .
Proof. We can suppose that the case is similar). Then by the mean value theorem we can find such that . Let . Since is continuous we get and . It follows from the definition of and the mean value theorem that
in the sense that there exists a constant such that for every we have .
Since belongs to we get Thus Following (18) and (21) we get Since for all we get Thus thanks to Proposition 6 we have which yields .
By replacing by we also have the following result.
Lemma 11. Let be in and the domain below the graph of . Consider . Suppose that there exist , as , , and , such that Then .
3.2. Dyadic Expansions and Approximation by Dyadics
We give some properties of the approximation of a point by the dyadics. Such properties will be used later.
Let . Set , the binary digits of ; that is, (i)Note that dyadic points, that is, points with , are characterized by the fact that one can find such that and for or equivalently and for . Furthermore for the number is dyadic. Since , then . On the other hand has the simple expansion . We will denote by the set of all dyadic points in .(ii)Let us come back to the general case with any point in . For each , define () by Set Define the rate of approximation of by dyadics as Since , then, for every , we have . If is dyadic then (by taking the convention ). If is normal (i.e., the frequency of ones (or zeros) in the binary expansion of is equal to ) then .(iii)If , following the definition of , then for any such that one can find a subsequence for such that Let . We then have Thus, either belongs to the dyadic interval and in this case it satisfies , or it belongs to the other interval and in this case it satisfies . In both cases let us notice that the binary expansion of contains chains of or whose length increases when .
3.3. Approximation by Sequences of Maxima of
We will see in the following that points in of the set will play a big role in this work, since they actually are the locations of the local maxima of the function (see below). Remark that they are characterized by the fact that, for each , one can find such that for we have .
As in the case of dyadic approximation we can define a rate of approximation by this kind of points.
Indeed let for Define Then the rate of approximation of by elements of is given by Since , then, for every , we have .
In the case of dyadic numbers, we have . But remark that in other nontrivial cases there is no obvious relationship between and . Indeed one can check on the following examples that and can independently take any value.(i)Let with and for all . Then we have .(ii)Let . Then with the integer part of . We have whereas .(iii)Let . Then . We have whereas .(iv)Let and . Let . Then and .
3.4. The Shift Operator
Since it is easy to check that we obtain from (2) with and The term of (35) corresponding to is . But, the function is supported in , and therefore vanishes outside and for For dyadic rationals , with , as we already said it, there exist two binary expansions, one such that and for and another one such that and for . The two right-hand sides of (36) corresponding to the two choices of give identical results.
Denote by the shift operator Observe that Hence
Our self-similar function is of the form with
For , denote Remark that is affine on intervals of type . Remark also that if , then . So, if and , then .
It follows that if then the slope of at any point of the interval is exactly
3.5. Extrema of
Proposition 12. Let and let be the function defined by (39); then (i)0 and 1 are the abscissas of the global minima of ;(ii)the dyadic points are the abscissas of the minima of and furthermore with .
In the case of the maxima, the statement of the result is slightly more technical. We need the following proposition of  using the same notations used previously.
Proposition 13. Let and let be the function defined by (39). Let and the list of positions where attains its maximum on . Let be the maximum on of . Then (i) for .(ii) if .(iii) for all .(iv).
The following proposition is a consequence of the previous one.
Proposition 14. Let and let be the function defined by (39). Then (i) and are the abscissas of the global maxima of ;(ii)the abscissas of the local maxima of are the points of .
3.6. Approximation of Slopes of
Suppose first we have some information about the dyadic expansion of . Then we have the following lemma, which helps to control the behavior of the slopes of the affine function .
Lemma 15. Let be a dyadic number. Then one can find , , and depending only on such that if then
Let be the abscissa of a local maximum of . Then one can find , , and such that for
Let be a nondyadic point such that . Then one can find two subsequences and with for all , such that and for . Furthermore one can find , , and such that for
Let be a nondyadic point such that . Then one can find two subsequences and with for all , such that for and . Furthermore one can find , , and such that for
Proof. Consider the following cases.
Case 1. The idea is very simple since it is a direct computation.
Indeed following (43) we have for The second equation with can be computed in the same way, up to a change of signs.
Thus one can find , , and such that (46) holds for .
Case 2. It is enough to remark that has the following binary expansion: As a consequence of Proposition 14, the same kind of computation yields Case 2.
Case 3. Since , for any one can find two subsequences and such that and with .
Suppose first eventually up to a small change of definition of that and .
Then with the same kind of computation as in Case 1 one gets In the other case , the sign of the slope will be changed.
Case 4. This follows exactly the same ideas than previously. Since for any one can find two subsequences and such that for all and , . Then with the same kind of computation as in Case 1 one gets Hence Lemma 15 holds.
If we do not have any further information on , the following lemma will be useful.
Lemma 16. Let be a nondyadic number. (1)Then there exists and such that for all one can find such that (2)If then there exists such that for all there exists such that (54) holds and(i)either and ,(ii)or and .
Proof. (1) The upper bound is a straightforward computation.
Suppose the contrary; that is, for all one can find such that for all If we suppose without loss of generality that then at step It is enough to choose such that to have a contradiction.
(2) Suppose the contrary; that is, there exists and that, for all , there exists , such that for all (a)either ,(b)or ,(c)or .
Remark first that the points of satisfy exactly (2b) and (2c). Indeed for and assuming that , has a binary expansion (51).
Thus following (43) the slope satisfies Hence, for being large enough (2b) and (2c) are satisfied.
Our goal is thus to prove that if we choose being small enough then only (2b) and (2c) can be satisfied, which will lead to the fact that and thus to a contradiction.
Let us start by the following special cases.
(i) We claim that if one can find being large enough such that then , which is a contradiction.
Let us prove this claim.
We will need the following sequence: let for We clearly have for all .
Choose and such that the hypotheses are satisfied.
Suppose that is such that . Then .
Remark that and thus .
Suppose without loss of generality that (the case is symetrical and can be proved in exactly the same way). Thus and Let us prove by induction on that for all We just proved that is true for .
Suppose that for is true. Suppose without loss of generality that (the case is symetrical and can be proved in exactly the same way). Thus and Since satisfies exactly , we have the result and property is satisfied at level .
Thus for all is true. Recall that since for all , this implies that for all , which is exactly the characterization of the points in and is in contradiction with the hypothesis .
In the following we will always keep the hypothesis so that for large enough we always have .
(ii) We now consider the case where is close to the value of and prove that this yields that , thus a contradiction.
Let and whose value will be precised later on.
Suppose is such that (the case can be done exactly in the same way). Then and ; hence Choose such that ; hence , which is possible since .
This yields . Thus Let us prove by induction on that, for all , We just prove that the case is true.
Suppose one can find such that is true for all .
Let us prove that it is true at . Without loss of generality suppose ; thus .
We have This proves that is true at .
Thus by induction is true for all . This means that for all , and thus , hence the contradiction.
(iii) We now study the case where and prove that if we choose being small enough then it will lead to .
Indeed let with And suppose such that . Suppose without lost of generality. Thus we have Remark that with the choice of we made, we have on one hand and on the other hand