Differentiability of a two-parameter family of self-affine functions
This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called -expansions) and the infinite derivatives of a class of self-affine functions. Precisely, we extend Okamoto’s function (itself a generalization of the well-known functions of Perkins and Katsuura) to a two-parameter family . We first show that for each , is either , , or undefined. We then extend Okamoto’s theorem by proving that for each , depending on the value of relative to a pair of thresholds, the set is either empty, uncountable but Lebesgue null, or of full Lebesgue measure. We compute its Hausdorff dimension in the second case.
The second result is a characterization of the set , which enables us to closely relate this set to the set of points which have a unique expansion in the (typically noninteger) base . Recent advances in the theory of -expansions are then used to determine the cardinality and Hausdorff dimension of , which depends qualitatively on the value of relative to a second pair of thresholds.
AMS 2010 subject classification: 26A27 (primary); 28A78, 11A63 (secondary)
Key words and phrases: Continuous nowhere differentiable function; infinite derivative; beta-expansion; Hausdorff dimension; Komornik-Loreti constant; Thue-Morse sequence.
The aim of this paper is to investigate the differentiability of a two-parameter family of self-affine functions, constructed as follows. Fix a positive integer and a real parameter satisfying , and let be the number such that . Note that . Let , , and for , put , and . Now set , and for , define recursively on each interval () by
Each is a continuous, piecewise linear function from the interval onto itself, and it is easy to see that the sequence converges uniformly to a limit function which we denote by . This function is again continuous and maps onto itself. It may be viewed as the self-affine function “generated” by the piecewise linear function with interpolation points , . When we have Okamoto’s family of self-affine functions , which includes Perkins’ function  for and the Katsuura function  for ; see also Bourbaki . Figure 1 illustrates the above construction for ; graphs of for two values of are shown in Figure 2; and Figure 3 illustrates the case .
The restriction is not necessary; when we have and is a generalized Cantor function. When , we have and is strictly singular. Since the differentiability of such functions has been well-studied (e.g. [6, 7, 8, 10, 12, 15, 26]), we will focus exclusively on the case , when is of unbounded variation. However, see  for a detailed analysis of the case , .
Our main goal is to study the finite and infinite derivatives of , thereby extending results of Okamoto  and Allaart . We first show that for each the differentiability of follows the trichotomy discovered by Okamoto  for the case : There are thresholds and (depending on ) such that is nowhere differentiable for ; nondifferentiable almost everywhere for ; and differentiable almost everywhere for . We moreover compute the Hausdorff dimension of the exceptional sets implicit in the above statement.
In  a surprising connection was found between the infinite derivatives of and expansions of real numbers in noninteger bases. Our aim here is to generalize this result. We first give an explicit description of the set of points for which , and then show that this set is closely related to the set of real numbers which have a unique expansion in base over the alphabet , where . This allows us to express the Hausdorff dimension of directly in terms of the dimension of , which is known to vary continuously with and can be calculated explicitly for many values of ; see [17, 20]. We also take advantage of other recent breakthroughs in the theory of -expansions to obtain a complete classification of the cardinality of .
To end this introduction, we point out that the box-counting dimension of the graph of is given by
This follows easily from the self-affine structure of , for instance by using Example 11.4 of Falconer . The Hausdorff dimension, on the other hand, does not appear to be known for any value of , even when ; see the remark in the introduction of .
2 Main results
2.1 Finite derivatives
Following standard convention, we consider a function to be differentiable at a point if has a well-defined finite derivative at . For each , let , let
and let be the unique root in of the polynomial equation
If , then is nowhere differentiable;
If , then is nondifferentiable almost everywhere, but is differentiable at uncountably many points;
If , then is differentiable almost everywhere, but is nondifferentiable at uncountably many points.
Moreover, if is differentiable at a point , then .
Statements (ii) and (iii) of the above theorem involve uncountably large sets of Lebesgue measure zero; it is of interest to determine their Hausdorff dimension. Let
Define the functions
For a set , we denote the Hausdorff dimension of by .
If , then ;
If , then ;
The function is concave on , takes on its maximum value of at , and its limits as and as are and , respectively.
Observe that for , has a discontinuity at , where it jumps from to ; see Figure 4.
2.2 Infinite derivatives
For , let
denote the expansion of in base , so for each . When has two such expansions, we take the one ending in all zeros. Let
We also write , and note that when is even, . For , write .
Let . A point satisfies if and only if and the following two limits hold:
Assuming all these conditions are satisfied, if is even, and if is odd.
While the conditions (4) and (5) may look complicated at first sight, readers familiar with -expansions will recognize the summations appearing in them. For a real number and , we call an expression of the form
an expansion of in base over the alphabet (or simply, a -expansion). Clearly such an expansion exists if and only if . It is well known (see ) that almost every in this interval has a continuum of -expansions. For the purpose of this article, we reduce the interval a bit further and consider the so-called univoque set
Let . For , let denote the projection map given by
so that (6) can be written compactly as . Let denote the left shift map on ; that is, . Define the set
and this, together with Theorem 2.3, suggests a close connection between the set
and the univoque set , where . The size of has been well-studied, starting with the remarkable theorem of Glendinning and Sidorov . There are two pertinent thresholds, which we now define. First, for , let
Baker  calls a generalized golden ratio, because .
Next, recall that the Thue-Morse sequence is the sequence of ’s and ’s given by , where is the number of ’s in the binary representation of . Thus,
For each , define a generalized Thue-Morse sequence by
The set is:
empty if ;
nonempty but countable if ;
uncountable but of Hausdorff dimension zero if ;
of positive Hausdorff dimension if .
(In case (ii), there is a further threshold between and that separates finite and infinite cardinalities of , but this is not relevant to our present aims.)
Now let , and . For a finite set , let denote the set of all finite sequences of elements of , including the empty sequence.
For all and for almost all ,
where and denotes the concatenation of with ;
For all , is a subset of the set in the right-hand-side of (7), and the inclusion is proper for infinitely many , including itself;
For all ,
Theorem 2.5(i) says that for almost all , the set consists precisely of those points whose base expansion is obtained by taking an arbitrary point having a unique expansion in base (where ), doubling the base digits of , and appending the resulting sequence to an arbitrary finite prefix of digits from .
The set is:
empty if ;
countably infinite, containing only rational points, if ;
uncountable with Hausdorff dimension zero if ;
of strictly positive Hausdorff dimension if .
Moreover, on the interval , the function is continuous and nonincreasing, and its points of decrease form a set of Lebesgue measure zero.
Regarding the relative ordering and the asymptotics of the five thresholds in Table 1, we have the following:
For each , we have
As , we have
It is interesting to observe that for , there is an interval of -values (namely, ) for which but . In other words, for such there are uncountably many points where is differentiable, but no points where it has an infinite derivative. For there is no such , but there is still an interval (namely, ) for which but is only countable. For all , however, whenever .
Recall that for , denotes the expansion of in base . We first introduce some additional notation. For , let denote the number of odd digits among . Let , and put for and . For and , let denote that interval which contains .
The first important observation is that
Next, recall that is the number such that . The recursive construction of the piecewise linear approximants implies that
at all not of the form , . As a result,
and since neither of these two values equals , cannot converge to a nonzero finite number. Clearly, if exists, it must be equal to in view of (10). The only possible finite value of , therefore, is zero.
For , if and only if .
Only the “if” part requires proof. For simplicity, write . Let denote the slope of on the interval . An easy induction argument shows that
Now assume . Fix , let be the integer such that , and let be such that . Then and . If , (13) gives
where , and the last inequality follows from (12). If , the same bound follows even more directly. Thus, we obtain the estimate
showing that has a vanishing right derivative at . By a similar argument, has a vanishing left derivative at as well, and hence, . ∎
and use (11) to write
The significance of the function is that
Since , this last equation together with Lemma 3.1 implies
Proof of Theorem 2.1.
(i) Assume first that . Since , (11) yields
for every not of the form , so for such . Hence, is nowhere differentiable.
(ii) and (iii): By Borel’s normal number theorem,
By definition of , we have . Moreover, is monotone increasing on . From these observations, it follows via (14) and (15) that has Lebesgue measure one if , and Lebesgue measure zero if . Finally, the law of the iterated logarithm implies that for almost every , , and therefore , for infinitely many . (See  for more details in the case ). Thus, has measure zero as well. The remaining statements follow from Theorem 2.2, which is proved below. ∎
In view of the relations (14), we define for the sets
It is well known (see, for instance, [9, Proposition 10.1]) that
If , then has Lebesgue measure one by (15). Suppose . Then contains the set
which has Hausdorff dimension by (18). Therefore, .
For the reverse inequality, we introduce a probability measure on as follows. Set
This defines for all and in such a way that for all and , and hence extends uniquely to a Borel probability measure on , which we again denote by . It is a routine exercise that concentrates its mass on the set , so in particular . It now follows just as in the proof of [1, Lemma 4.2] that if , then
where denotes the length of . Using [9, Proposition 4.9], we conclude that . ∎
To avoid notational clutter we again write . In order for to have an infinite derivative at , it is clear that must tend to . By (11), this is the case if and only if is even for all but finitely many . However, it turns out that this condition is not sufficient.
We begin with an infinite-series representation of (see  for a proof when ):
where are the numbers used in the introduction to define . In the special case when is even for every , this reduces to
where , and we have used that for . Let and denote the right-hand and left-hand derivative of , respectively.
Assume is even for every , and let . Then if and only if
For each , let be the integer such that , and put (the right endpoint of ). In order that , it is clearly necessary that
The slope of