DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2017017 DYNAMICAL SYSTEMS SERIES S Volume 10, Number 2, April 2017 pp. 353–365

DERIVATIVES OF SLIPPERY DEVIL’S STAIRCASES

JUN-JIE MIAO Department of , Shanghai Key Laboratory of PMMP East China Normal University, Shanghai 200241, China

SARA MUNDAY Department of Mathematics, University of Bologna Bologna, Italy

Dedicated to the memory of Bernd O. Stratmann

ABSTRACT. In this paper we first give a survey of known results on the of slippery Devil’s staircase functions, that is, functions that are singular with respect to the Lebesgue and strictly increasing. The best known example of such a is the Minkowski question-mark function, which was proved to be singular by Salem, in a paper which introduced some other constructions of singular functions. We describe all of these examples. Also we consider various generalisations of the Minkowski question- mark function, such as α-Farey-Minkowski functions. These examples all arise from one- dimensional dynamics. A few open questions and suggestions for filling minor gaps in the literature are proposed. Finally, we go back to ordinary Devil’s staircases (i.e. non- decreasing singular functions) and discuss work done in that setting with the more general Hölder , and consider the outlook to extend those results to the strictly increasing situation.

1. Introduction and general preliminaries. Our first aim in this short note is to give a survey of results on topological conjugacy maps that are examples of slippery Devil’s staircases, which means that they are strictly increasing and singular, that is, they have derivative Lebesgue- equal to zero. In particular, we are interested in the exceptional set of points where the derivative of such a function is not equal to zero. It will quickly become clear, but let us say it immediately anyway, our late and sadly missed colleague Bernd Stratmann, along with his co-authors, made a very big contribution to this and related topics. In the last section we will discuss some results for ordinary Devil’s staircases, that is, singular functions that are increasing, but not strictly increasing. These functions are often found as distribution functions of measures on Cantor-type sets, and as such are obviously singular. The most well known example of such a function is Cantor’s ternary function, which arises from the distribution function of the self-similar measure on the Can- tor middle-third set. It was introduced by Cantor in [3], and, in the context we are interested in here, was studied by Darst [4], who initiated the dimension-theoretic study of points of nondifferentiability of Devil’s staircase functions. Darst found that the Hausdorff dimen- sion of the set of nondifferentiability points of Cantor’s ternary function is (log2/log3)2.

2010 Mathematics Subject Classification. 37E05, 26A27, 28A80, 11K55. Key words and phrases. Survey, topological conjugacies, differentiability, , thermody- namical formalism.

353 354 JUN-JIE MIAO AND SARA MUNDAY

Let us recall that if we have two systems (X,T) and (Y,S), where all we assume is that the maps T and S are continuous maps on metric spaces, then they are topologically conjugate via a homeomorphism h : X → Y if we have that h ◦ T = S ◦ h. In other words, every orbit under the system T corresponds exactly to an orbit under the system S. Here, all the maps we will consider are maps of the unit interval with either finitely or countably many full inverse branches. Somewhat more precisely, they are all examples of what are sometimes called countable Markov maps, that is, there exist maps fi : [0,1] → [0,1], for i ∈ A ⊆ N, which are continuous and monotonic on [0,1] and differentiable on (0,1), and S we assume that i∈A fi([0,1]) = (0,1] and if i 6= j then fi((0,1))∩ f j((0,1)) = ∅. Then the Markov map T is defined to be  f −1(x), if x ∈ f ([0,1)); T(x) := i i 0, if x = 0. With such functions it is convenient to use the fact that they are topologically conjugate to the left-shift map on the alphabet A. (Strictly speaking, we must exclude a countable set of points which have two possible codings.) So, we can think either of points in the unit interval acted upon by T, or points in the sequence space AN acted upon by the left-shift map σ, where σ(ω1,ω2,ω3,...) = (ω2,ω3,ω4,...). The relation between these two systems comes from the projection map πT : (AN,σ) → ([0,1],T) which is given by

πT (ω1,ω2,...) := lim fω ◦ fω ◦ ··· ◦ fω (0). n→∞ 1 2 n Note that this limit always exists (by Cantor’s intersection theorem). We will use the notion of a (level n) cylinder set, which means the set of all points whose coding coincides up to the first n points, i.e., C(x1,...,xn) := {(y1,y2,y3,...) : xi = yi for all 1 ≤ i ≤ n}. The paper is organised as follows. Section 2 is devoted to perhaps the most classical example of a , namely, Minkowski’s question-mark function. This has been investigated by several authors, and we give references below, but it was in particular shown to be singular in a 1943 paper of Salem [22]. In Section 3, we describe the other ex- amples of singular functions given by Salem in the same paper, and add some observations and open questions. In Section 4, we collect various other related examples that can be found in the literature, some of them direct generalisations of Minkowski’s question-mark function, some of them related in the way they behave rather than the way they are defined. Finally in Section 5, instead of the usual derivative, we will make some remarks about the q-Hölder derivatives of certain functions. As already mentioned above, in this last section the existing results we mention are all for ordinary Devil’s staircase functions.

2. Minkowski’s question-mark function. Minkowski’s question-mark function, which we shall denote by Q : [0,1] → [0,1], was originally introduced by Minkowski [18] and later investigated by Denjoy [5] and Salem [22], amongst others. Minkowski’s original motivation behind the definition of the function that now bears his name was to highlight a certain intriguing property of continued fractions, namely that the set of quadratic surds corresponds precisely to the set of real numbers that admit an eventually-periodic continued fraction expansion. In other words, if x ∈ [0,1] can be written as a continued fraction of the form [x1,...,xm,xm+1,...,xm+k], then x is an irrational root of some quadratic polynomial with integer coefficients and, moreover, the converse statement also holds. This result is sometimes referred to as Lagrange’s Theorem [16]. Minkowski designed the function Q to map the quadratic surds into the non-dyadic rational numbers in a continuous and order- preserving way. The question-mark function is constructed in the following way. First, DERIVATIVES OF SLIPPERY DEVIL’S STAIRCASES 355

1 1

7/8

3/4

5/8

1/2

3/8

1/4

1/8

0 0 1/41/3 2/51/2 3/52/3 3/4 1 0 1

FIGURE 2.1. On the left, the functions Q1, Q2 and Q3, and on the right, the Minkowski question-mark function, Q : [0,1] → [0,1]. define Q(0) = Q(0/1) .= 0 and Q(1) = Q(1/1) .= 1. Then, define  p + p0  Q(p/q) + Q(p0/q0) Q .= . q + q0 2 In other words, the function Q is successively defined on all the rational numbers in the unit interval by taking mediants of those that have already been defined, where the mediant of two fractions a/b and c/d is defined to be (a + c)/(b + d). The definition of Q is extended to all of [0,1] by continuity. Another way to think about the question-mark function is as a uniform limit of the sequence of piecewise linear functions (Qn)n∈N, where each Qn : [0,1] → [0,1] is defined by mapping the n-th level Stern-Brocot fractions, arranged in increasing order, onto the set {p/2n : 0 ≤ p ≤ 2n} and then joining these image points by straight line segments. The Stern-Brocot fractions are defined by starting at level 0 with with the set {0/1,1/1} and successively taking mediants. Levels one to three of these sets are as follows: 0 1 1 0 1 1 2 1 0 1 1 2 1 3 2 3 1 , , , , , , , , , , , , , , , , ,... 1 2 1 1 3 2 3 1 1 4 3 5 2 5 3 4 1 For a more detailed description of the Stern-Brocot sequence, we refer to [13] and refer- ences therein. Above, in Figure 2.1, can be found an illustration of the first few of the functions Qn and also an approximation of the graph of Q itself. Denjoy (and later Salem) demonstrated that the function Q is given by the following formula: ∞ k k ∑ xi Q([x1,x2,x3,...]) = −2 ∑(−1) 2 i=1 . k=1 Salem also derived the most important properties of Q from this formula, including the facts that Q is strictly increasing and singular with respect to , λ. Recalling from the introduction, this means that λ x ∈ [0,1] : Q0(x) = 0 = 1. Given this result, finer questions about the derivative can be asked. It was proved first in [20], and then in a far simpler way in [14], that if the derivative of Q at a point x exists, or is infinite, then Q0(x) ∈ {0,∞}. 356 JUN-JIE MIAO AND SARA MUNDAY

In other words, the unit interval can be split into the disjoint union of the three sets 0 0 Λ0 := {x ∈ [0,1] : Q (x) = 0}, Λ∞ := {x ∈ [0,1] : Q (x) = ∞} and 0 Λ∼ := {x ∈ [0,1] : Q (x) neither exists nor equals infinity}. With this in mind, there have been two strands of further research. One is to give de- scriptions of the points which lie in each of the sets Λ0 and Λ∞ in terms of their continued fraction expansions. Results of this type can be found in [20], and then improved upon in the papers [6, 7]. The second, which we will focus on here, is given by Kesseböhmer and Stratmann [14], who calculate explicitly the Hausdorff dimension of the sets Λ∞ and Λ∼ in terms of their previous thermodynamical results [13]. The analysis of the derivative of Q begins with the observation that Q also coincides with the topological conjugacy map between the Farey map, F : [0,1] → [0,1] and the tent map T : [0,1] → [0,1], where  1  1 x/(1 − x) for 0 ≤ x ≤ 2 ; 2x, for 0 ≤ x ≤ 2 ; F(x) := 1 and T(x) := 1 (1 − x)/x for 2 < x ≤ 1. 2 − 2x, for 2 < x ≤ 1. This allows one to think of Q as the map that sends cylinder sets coded by the Farey map onto cylinder sets coded by the tent map (note that each of these in level n has size 2−n), and in fact the proof that the derivative of Q can be either 0 or infinite is based on the preliminary (and easily established) result that if the derivative Q0(x) exists or is infinite at a point x, then 2−n Q0(x) := lim , n→∞ λ(Tn(x)) where here Tn(x) refers to the unique Farey cylinder set containing the point x and λ denotes the Lebesgue measure. This observation led Kesseböhmer and Stratmann to the idea that the Hausdorff dimension of the sets Λ∼ and Λ∞ can be related to the fine spectrum of the Lebesgue measure, given by the Hausdorff dimension of the level sets L (s),   log(λ(Tn(x))) L (s) := x ∈ [0,1] : lim = s , n→∞ −n

Then, where dimH denotes the Hausdorff dimension, the main result of [14] is as follows:

dimH(Λ∼) = dimH(Λ∞) = dimH(L (log2)) < dimH(Λ0) = 1. We note that log2 is the topological entropy of the maps F and T.

3. Salem’s examples. 3.1. Salem’s examples. As we have already noted, Salem [22] proved the singularity of the function Q. In the same paper, he also gave two further classes of examples, which we now describe. The first will turn out to be a special case of the type of function considered in [10], of which we will see more later, but the second seems to us to be still rather underdeveloped. The first of Salem’s examples is a function built as a uniform limit of piecewise linear functions, similarly to the definition of the Minkowski question-mark function. We start with a non-trivial probability vector p := (p0, p1), that is, p0 + p1 = 1 with p0, p1 > 0. Also, to avoid trivialities later, suppose that p0 6= 1/2. Then consider the straight line segment in the plane joining two points P(x,y) and Q(x + ∆x,y + ∆y). Let R be the point with co-ordinates (x + ∆x/2,y + p0∆y). If we replace the segment PQ with the two joined segments PR and RQ, we will say that we have performed the transformation T(p0, p1) on PQ. DERIVATIVES OF SLIPPERY DEVIL’S STAIRCASES 357

So, to define the desired function, start with fp,0(x) := x on [0,1] and apply T(p0, p1) to its graph. Then the result represents a function fp,1(x). Proceeding in this way, in the n n-th step we obtain a function fp,n : [0,1] → [0,1] consisting of 2 line segments. The sequence ( f ) then converges uniformly to a strictly increasing p,n n∈N fp : [0,1] → [0,1]. We can obtain this function in other ways. It is easily seen to be equivalent to define fp as the topological conjugacy between the functions ( x  , for x ∈ [0, p0); 2x, for x ∈ [0,1/2); p0 T2(x) := and Tp(x) := x−p0 2x − 1, for x ∈ [1/2,1]. , for x ∈ [p0,1] . p1

Alternatively, fp coincides with the distribution function of the (p0, p1)-Bernoulli measure on the map T2. Salem proved that each of these functions are singular (his proof utilises the fact that Lebesgue-almost every x ∈ [0,1] is normal in base 2), and moreover shows that fp is Hölder continuous with exponent log(max{p0, p1})/log2. As to finer questions about the deriva- tive, we will address this in the next section, since these examples, as part of a larger class containing them, have been studied subsequently in some detail. So, let us now describe the second of Salem’s examples. Instead of starting with a single vector (p0, p1), now we choose a sequence p := ((p0,n, p1,n))n≥1 such that p0,n and p1,n satisfy the same conditions for each n ∈ N that (p0, p1) did in the first example, and build a function Fp in a similar way. Again we begin with the straight line segment joining the origin to the point (1,1). To obtain the first approximating function Fp,1, we apply the transformation T(p0,1, p1,1) to this segment. Then, we get Fp,2 by performing T(p0,2, p1,2) on both segments of Fp,1. Proceeding thus, to obtain Fp,n, we apply T(p0,n, p1,n) to each n−1 of the 2 segments of Fp,(n−1). This construction, since we do the same thing to each subinterval in each layer, is an example of a homogeneous construction. Now, set

rn := 1 − 2p0,n.

If −α < rn < α for some α ∈ (0,1) and for every n ∈ N, then the above sequence of func- 1 tions again converges uniformly to a strictly increasing continuous function Fp : [0,1] → [0,1]. So let us assume that Fp is strictly increasing and continuous. Then, the main result concerning the functions Fp in [22] can be stated as follows: ∞ 2 The function Fp is singular if and only if the series ∑ rn diverges. n=1 There are then a number of natural questions that suggest themselves. For instance: • Is there some good way of quantifying, given a random choice of sequence p = ((p0,n, p1,n))n≥1, how likely it is that the function Fp is singular? • Can Fp be usefully described as a topological conjugacy or distribution function? • Supposing that Fp is singular, what are the values the derivative can take? • Again supposing Fp is singular, what about finer Hausdorff dimension questions about the derivative? At least the third question we can answer immediately. 0 Proposition 3.1. Let Fp be defined as above, and suppose that it is singular. Then, if Fp(x) exists or is equal to infinity, 0 Fp(x) ∈ {0,∞}.

1This condition is stronger than necessary, but it at least has the virtue of being easy to state. 358 JUN-JIE MIAO AND SARA MUNDAY

0 Proof. Suppose that we have x ∈ [0,1] such that Fp(x) exists or equals infinity. Write x = [x1,x2,...], where this is the binary representation of x, given by the map T2, so each xi ∈ {0,1}. Then the first step of the proof is to notice that λ(I (x)) F0(x) = lim n , (3.1) p n→∞ 2−n where In(x) is the n-th level cylinder set with size px1,1 px2,2 ... pxn,n. This can be seen by essentially copying the proof of Lemma 3.1 from [19]. So, by way of contradiction, suppose that there exist y ∈ [0,1] and 0 < c < ∞ with 0 Fp(y) = c. Then, in light of (3.1), we have that λ(I (y)) 1 lim n+1 = , n→∞ λ(In(y)) 2 which implies immediately that 1 lim py ,n+1 = . n→∞ n+1 2 This is equivalent, since for every n we have that p0,n + p1,n = 1, to saying that 1 lim p0,n+1 = . (3.2) n→∞ 2 There are then two cases to consider. The first is straightforward: If limn→∞ p0,n+1 6= 1/2, we are finished. So, for the second, suppose that we are in the situation that (3.2) holds. Then, comparing that to (3.1) again, we see that if y exists as above, we must have 0 that for every x such that Fp(x) exists that λ(I (x)) lim n = F0(x) > 0, n→∞ 2−n p since the sequences in the denominator and numerator must tend to zero at the same rate. But this contradicts the assumed singularity, so no such y exists. Remark 3.2. We have that for Minkowski’s question-mark function, Salem’s examples and all the other examples we will encounter in the next section, that whenever they are singular, the derivative is either 0 or infinite. One might expect that this is always the case for any singular function. However, it is not, as was shown (for example) in [23]. On the other hand, the example in there is built by gluing another family of singular functions, whose definitions have a relation to the ternary expansion of real numbers, into a function defined with help of the Cantor middle-third set. The explicit construction is done in several steps and it is not, at least on the face of it, possible to represent it as a topological conjugacy map.

4. Further examples. In this section, we shall collect various other examples of strictly increasing singular functions available in the literature, in more-or-less chronological order. 4.1. Expanding Markov maps. The class of examples we come to now is more general than those given by Salem. In [10], Jordan et al. consider topological conjugacies between what they call expanding piecewise C1+ε maps. Briefly, these are expanding maps of the unit interval which have d increasing full inverse branches, for some integer d ≥ 2, and each of these inverse branches is a strictly contracting C1+ε diffeomorphism on [0,1], for a fixed ε > 0. Each of these maps is a factor of the full shift on the alphabet A := {1,...,d}, DERIVATIVES OF SLIPPERY DEVIL’S STAIRCASES 359 since there are a countable set of points which have two possible codings. If S and T are two such maps, then they are topologically conjugate to each other, via a map θ := θ(S,T). In [10], a detailed fractal analysis of the following sets is given: 0 0 D0 = D0(S,T) := {x ∈ [0,1] : θ (x) = 0}, D∞ = D∞(S,T) := {x ∈ [0,1] : θ (x) = ∞} and 0 D∼ = D∼(S,T) := {x ∈ [0,1] : θ (x) neither exists nor equals infinity}. We will give some further details for this case (and for a similar one below). The main tool used to investigate the above sets is the thermodynamical formalism on the symbolic space (AN,σ), where A := {1,...,d}. So, let us now summarise very briefly the (by now) standard results we will need. We will only outline the case of a finite alphabet here, and for this more details can be found in, e.g., [21], but for the countable situation, the interested reader is referred to Mauldin and Urbanski´ [17] and the series of works by Sarig, see for instance [24, 25]. Throughout, continuous functions g : AN → R will be called potentials, and we use the n−1 k standard notation Sng := ∑k=0 g◦σ for Birkhoff (or ergodic) sums. The first definition we need is that of the pressure of a potential g: 1 P(g) := lim log exp( sup Sng(x)). n→∞ ∑ n ω∈An x∈[ω]

Let Mσ denote the collection of σ-invariant measures, that is, all measures µ such that µ ◦σ −1 = µ. A very deep and useful result which relates the pressure function to the metric (Kolmogorov-Sinai) entropy h(µ) of measures µ ∈ Mσ is the Variational Principle, which states: for any continuous g : AN → R we have that  Z  P(g) = sup h(µ) + gdµ : µ ∈ Mσ .

If a measure µ attains this supremum, we say that µ is an equilibrium measure for g. It is well known that in the case of a finite alphabet, unique equilibrium measures always exist for Hölder continuous potentials (and, moreover, they are Gibbs measures). These results can be found in Appendix II of [21]. The canonical, Hölder continuous geometric potentials ϕ,ψ : AN → R<0 for S and T, the maps conjugated to each other, are given for x = (x1,x2,...) ∈ AN by (x) := log|(S−1)0( ( (x)))| and (x) := log|(T −1)0( ( (x)))|, ϕ x1 πS σ ψ x1 πT σ where S−1 denotes the inverse branch of S associated to the letter x ; similarly for T. (Note x1 1 that when the branches are all positive, there is no need for the absolute value in the above definition, but later we will let one of the slopes be negative.) There then exists a function β : R → R given by the pressure equation P(sϕ + β(s)ψ) = 0, for all s ∈ R.

This function is well defined, since ψ < 0. Then, if µs denotes the equilibrium measure associated with the potential sϕ + β(s)ψ, R 0 − ϕ dµs β (s) = R < 0. ψ dµs Thus, β is a strictly decreasing function; moreover β(1) = 0 and β(0) = 1. If S and T are cohomologically independent, which means that there are no non-trivial choices of b,c ∈ R and continuous function u such that bϕ + cψ = u ◦ σ − u, then β turns out to be strictly 0 convex and (by the mean value theorem) there exists a unique s0 ∈ (0,1) with β (s0) = −1. 360 JUN-JIE MIAO AND SARA MUNDAY

Level sets L (s) here are defined to be ( −1 ) Snϕ(π (x)) L (s) := x ∈ [0,1] : lim S = s , n→∞ −1 Snψ(πT (x)) and (see [21] again), we have that

dimH(L (s)) = inf(t + β(t)/s). t∈R In other words, the dimension of the level set is given by the Legendre transformation βb(s)/s. One of the main results in [10] is then as follows: If S and T are two cohomologically independent expanding piecewise C1+ε maps, then the conjugacy map θ between them is singular and

0 < dimH(D∼) = dimH(D∞) = dimH(L (1)) = β(s0) + s0 < 1. Remark 4.1. 1. Observe here that if we were in the situation of maps with two branches and the map S was given by S(x) = 2x (mod 1), then the function ψ above is just given by ψ(x) = −log2. Thus, in this case, the level set could also be written in terms of log2, as in the Minkowski question-mark function example. 2. The condition that the maps T and S should be cohomologically independent is im- portant. In [10] it is also shown that if T and S are instead cohomologically depen- dent, then the conjugacy map θ(T,S) is a C1+ε diffeomorphism, and hence absolutely continuous. The derivative in this case is everywhere positive and finite. The first basic strategy of the proof is to relate the actual derivative of the conjugacy map to a sort of symbolic version. There are certain difficulties here that essentially stem from the the fact that for a set of “end points” of cylinder sets, two points that are geometrically very close have symbolic codes that can differ by long blocks of 0s and 1s. Then the behaviour of the symbolic version of the derivative is investigated with relation to various equilibrium measures, for instance the lower bound for D∼ comes from analysing µs0 - typical points. The authors of [10] also investigate in detail two concrete examples in Section 4 of that paper, the first of which are the first of Salem’s examples, the maps fp described in Section 3. Here the Hausdorff dimension can be given explicitly: If we set

1 + log2(p0) 1 + log2(p1) P0 := and P1 := , −log2(p0) + log2(p1) −log2(p0) + log2(p1) then dimH (D∞) = dimH (D∼) = −P0 log2(P0) − P1 log2(P1). In part of the recent paper [11], a very similar situation is considered. Here, instead of taking conjugacies between expanding maps with d increasing branches, the set up is changed as follows: Suppose that we have two maps S,T : [0,1] → [0,1] which have exactly two full branches, the left-hand one increasing and the right-hand decreasing, and both branches are strictly contracting C1+ε diffeomorphisms. We will refer to these as “tent- like” maps. Recall that C(x1,...,xk) denotes the (symbolic) cylinder set; we will denote the projec- tion of these sets to the interval by I(x1,...,xk). For the tent-like maps, the cylinder sets are k k arranged as follows: If (x1,...,xk) ∈ {0,1} is such that ∑i=1 xi is odd, then the two gener- ation k + 1 intervals contained in I(x1,...,xk), namely, I(x1,...,xk,0) and I(x1,...,xk,1), have I(x1,...,xk,0) on the left (thinking from the origin). If the sum of the digits is even, DERIVATIVES OF SLIPPERY DEVIL’S STAIRCASES 361 the k + 1 generation cylinders are the other way around, so I(x1,...,xk,0) is on the right. This means that if we have two level n sets such that I(x1,...,xn) ∩ I(y1,...,yn) 6= ∅, then if j := inf{1 ≤ k ≤ n : xk 6= yk}, we have, without loss of generality, that

(x1,...,xn) = (x1,...,x j−1,1,1,0,...,0) and (y1,...,yn) = (x1,...,x j−1,0,1,0,...,0). That is, the initial n-blocks of any two points in this intersection can only differ at the j-th co-ordinate. This is exactly the point where it is significantly easier to analyse these maps compared to those with increasing branches. More precisely, by considering estimates on the distance between points in neighbouring cylinders and estimates for the diameter of cylinder sets themselves in terms of ergodic sums of the geometric potentials ϕ and ψ, one can deduce the following key lemma: Fix x 6= y ∈ [0,1] and let n = inf{k : I(x1,...,xk) ∩ I(y1,...,yk) 6= ∅}. We have that |θ(x) − θ(y)|  eSn(ψ−ϕ)(x). (4.1) |x − y| Given this estimate, it is then straightforward to obtain, applying results from the ther- modynamical formalism as in [10], an exactly analogous result to that stated above for the sets of points where the derivative is infinite and those where it does not exist. This result can be found as Proposition 7.1 in [11]. Applying this to a “Salem-like” example, i.e., in- stead of the topological conjugacy between T2 and Tp we exchange the right-hand branches for decreasing ones with the same slope, the Hausdorff dimensions of the exceptional sets can be calculated exactly again and they are precisely the same as in the actual Salem case (see Theorem 1.6 in [11]). Of course, this is not surprising; it has been shown for both situations that the derivative can be investigated purely in terms of the shift on {0,1}N with the appropriate geometric potential functions, which are the same in both cases.

4.2. Generalisations of Q. Recall from Section 2 that Minkowski’s question-mark func- tion coincides with the topological conjugacy mapping between the Farey map and the tent map. One way to generalise this function is to exchange the Farey map for another map, which we will first choose from the family of α-Farey maps, which were introduced in [12]. Let us recall the definition. Denote by α := {An : n ∈ N} a countably infinite partition of the unit interval [0,1], consisting of non-empty, right-closed and left-open intervals, let ∞ an := λ(An) and finally let tn := ∑k=n ak. We assume that the elements of α are ordered from right to left, starting from A1, and that these elements accumulate only at the ori- gin. We can then define the α-Farey map, as follows. For a given partition α, the map Fα : [0,1] → [0,1] is given by   (1 − x)/a1 if x ∈ A1, Fα (x) := an−1(x −tn+1)/an +tn if x ∈ An, for n ≥ 2,  0 if x = 0.

The map Fα , in other words, maps the partition element A1 linearly onto [0,1) and maps An linearly onto An−1 for all n ≥ 2. For later use, let us also give the definition of the related class of α-Lüroth maps: For any given α as defined above, we have  (t − x)/a for x ∈ A , n ∈ ; L (x) := n n n N α 0 if x = 0. Remark 4.2. The relation here is that of jump transformation. In general, to define a jump transformation on a set A ⊂ [0,1] for a map T : [0,1] → [0,1], first we define the first passage time, ρ(x) := 1 + inf{n ≥ 0 : T n(x) ∈ A}, 362 JUN-JIE MIAO AND SARA MUNDAY and then say that the jump transformation of T on A is given by T ρ(x)(x) for all x ∈ [0,1]. It is a straightforward calculation to check that for each partition α, we have that Lα is the jump transformation of Fα on the set A1, the first element of the partition α. (Note that the same holds for the Gauss and Farey maps, that is, G is the jump transformation of F, in this case on the set [1/2,1].) Then, if T and S are maps conjugated by θ, then θ ◦ T n = Sn ◦ θ for all n ∈ N, and so θ is also the conjugacy map between the jump transformations of T on A and S on θ(A). The main result in [19] mirrors closely the result in [14]. First, consider the conjugacy map θα between an arbitrary map Fα and the tent map T (and note that the tent map is an α- −n −(n−1) Farey map for the dyadic partition αD := {(2 ,2 : n ∈ N}). Then for the derivative of θα , if it exists or is infinite, we have 0 θα (x) ∈ {0,∞}.

It can be deduced easily from this that each map θα is singular. To state the main result of [19], we must restrict the class of partitions used to those that are either expanding or expansive of exponent τ > 0. We recall the definitions from [12]: A partition α is −τ said to be expansive of exponent τ > 0 if for the tails of α we have that tn = n ψ(n), + for some slowly-varying2 function ψ : N → R , whereas α is said to be expanding if limn→∞ tn/tn+1 = κ, for some κ > 1. We then define the sets 0 0 Θ0 := {x ∈ [0,1] : θα (x) = 0}, Θ∞ := {x ∈ [0,1] : θα (x) = ∞} and 0 Θ∼ := {x ∈ [0,1] : θ (x) neither exists nor equals infinity}. Note that the singularity of θα implies also that dimH(Θ0) = 1. The main result of [19] is that dimH(Θ∞) = dimH(Θ∼) = dimH(L (log2)) < dimH(Θ0) = 1, where ( (α) ) log(λ(In (x))) L (s) := x ∈ [0,1] : lim = s , n→∞ −n (α) and here In (x) refers to the unique n-th level α-Farey cylinder set containing the point x. The Hausdorff dimension of these sets was calculated in [12]. Note that this is precisely the same as the analogous result outlined in Section 2 for the Minkowski question-mark function. It ought to follow reasonably easily from the techniques in [11] that an analogous result can be shown to hold for a wider class of partitions, and for conjugacies also between any two distinct α-Farey maps defined by partitions from this class (i.e., we do not need one of the maps to be the tent map). That is, if instead of restricting ourselves to expanding and expansive partitions, we allow those with thin tails, as defined in [11], so there exists t < 1 with ∞ t ∑ ai < ∞, i=1 then by considering the conjugacy in question as being between α-Lüroth maps instead of α-Farey maps (which is equivalent, in light of Remark 4.2), and using the finite approx- imation property outlined in [11], it should to be possible to widen the result from [19]. Note that it is still possible to create partitions which violate this condition, so there will still be some particular examples of conjugacies between α-Lüroth maps for which another approach is needed.

2 + + A measurable function f : R → R is said to be slowly varying if limx→∞ f (xy)/ f (x) = 1, for all y > 0. DERIVATIVES OF SLIPPERY DEVIL’S STAIRCASES 363

Remark 4.3. Let us also remark that the main result in [11] has a slightly different flavour. There, it is shown that for a wide class of countable Markov maps (with the thin tail con- dition stated above and also a requirement for summable variations), if Tk is a sequence of maps that converges pointwise to a map T, then the conjugacy maps θk between Tk and T 0 have the property that dimH({x ∈ [0,1] : θk(x) 6= 0}) → 1 as k tends to infinity. Another recent example is given by Arroyo [1]. In this paper, the author considers a different generalisation of Minkowski’s question-mark function. Instead of the conjugacy between the Gauss map and the αD-Lüroth map, he allows the Gauss map to be conjugated to any arbitrary α-Lüroth map, and denotes the resulting conjugacy map by ?α : [0,1] → [0,1], where for x = [x1,x2,x3,...] given as a continued fraction expansion, we have

n−1 n+1 ?α (x) := tx1 + ∑ (−1) ∏ ax j txn = [x1,x2,x3,...]α . n=2 j=1 The main result in [1] is that all of these functions are singular. The proof is an elegant one, using the Gauss measure. However, the derivatives of these functions are not investigated in any greater detail. Remark 4.4. Fix ε > 0. Then there are plenty of examples of topological conjugacy 0 maps θ := θ(S,T) which satisfy dimH({x : θ (x) 6= 0}) > 1 − ε. Let us give one concrete construction, built using the tent map, which we recall can be thought of as an α-Farey map n n−1 defined using the dyadic partition αD := {(2 ,2 ] : n ∈ N} and a “perturbed version” of the tent map coming from an altered dyadic partition. To begin, for each N ∈ N, let us fix a partition αN such that the first N element match exactly the first N elements of αD, and the N + 1-th element is not equal to (2N+1,2N]. (The tail of the partition can be chosen arbitrarily for each N.) Now, let us define the sets

BN := {x = [`1(x),`2(x),...]αN ∈ [0,1] : `k(x) ≤ N for all k ∈ N}.

In other words, BN is the set of all those points in the unit interval whose αN-Lüroth expansions have only digits from the set {1,...,N}. It follows directly from Lemma 3.3 in x ∈ 0 (x) [19] that if BN then θαN does not exist. Furthermore, an application of Hutchinson’s Formula (see [9], Theorem 9.3), shows that

N −sN i dimH (BN) = sN, where sN is given by ∑ 2 = 1. i=1

The Hausdorff dimension of the sets BN, as given above, can be written in terms of the multinacci numbers. The N-th multinacci number mN is defined to be the unique solution of the equation xN + xN−1 + ··· + x − 1 = 0 that lies in (1/2,1) and it is easy to see3 that −sN limN→∞ mN = 1/2. Here, our x is equal to 2 , which implies that sN = log2(1/mN). Therefore, as N tends to infinity, we have that sN tends to 1. Thus, for any fixed ε > 0, we N s ≤ ( ) ≤ ({x 0 (x) 6= }) can find an sufficiently large that N dimH Θ∼ dimH : θαN 0 is at least equal to 1 − ε. With this in mind, consider the following question, this one proposed first by Thomas Jordan: • Can one find maps S and T, topologically conjugate via the map θ(S,T), that satisfy 0 dimH({x ∈ [0,1] : θ(S,T) (x) 6= 0}) = 1?

3Consider the left hand side of xN +xN−1 +···+x = 1. This tends to the geometric series with sum x/(1−x). 364 JUN-JIE MIAO AND SARA MUNDAY

5. q-Hölder derivatives. So far, we have only considered the ordinary derivative of the various examples of singular maps. In this final section, we will remark instead upon the q-Hölder derivatives of a function f : [0,1] → [0,1], which are defined for each q ∈ R by the following limit, provided that it exists:

| f (x) − f (y)| Dq f (x) := lim . (5.1) y→x |x − y|q

Kesseböhmer and Stratmann [15] considered this type of derivative for certain distribu- tion functions coming from Gibbs measures on conformal iterated function systems which satisfy the strong separation condition (in other words, there are gaps between the con- struction intervals). Where ψ denotes the potential used to generate the Gibbs measure and ϕ denotes the usual geometric potential, they needed the condition that ψ < ϕ. These functions are obviously singular, as they are all examples of ordinary Devil’s staircases, the most widely-known example being the Cantor ternary function, as mentioned already in the introduction. The paper [15] is a significant generalisation of previous works by various authors who had considered particular examples or classes of examples of these staircase functions. One particularly interesting case studied previously by Falconer [8] is that of Ahlfors-regular measures on self-similar sets. This can be considered a generalisation of the work of Darst [4] described in the introduction, as the “squaring property” he discov- ered for the Cantor ternary function can also be found here, in the sense that if Dq denotes the set of points where the distribution function of the Ahlfors-regular measure on the set E has no finite or infinite q-Hölder derivative, then

(dim (E))2 dim (Dq) = H . H q The advancement in [15] was achieved by the realisation that the natural tools to use for analysing such functions was the thermodynamical formalism, as in the cases outlined earlier. In a recent paper, Troscheit [26] was able to relax the assumption ψ < ϕ and thus obtain analogous results in a far more general setting. The question of extending this to the case of slippery Devil’s staircases seems to us to be interesting. In [10], Remark 1.6(4), it is stated that “straightforward adaptations” of the proofs there would yield the Hausdorff dimensions of the sets

(q) q D∞ := {x ∈ [0,1] : D θ(x) = ∞} and (q) q D∼ := {x ∈ [0,1] : D θ(x) neither exists nor equals infinity}. It is claimed that

(q) (q) dimH(D∞ ) = dimH(D∼ ) = dimH(L (s)), (5.2) where we recall that the level sets L (s) were defined in Section 4.1 above. In the first instance, given the slight technical difficulties noted for the case of conjugacies between interval maps with d increasing inverse branches, we suppose that to verify this claim ought to be rather easier in the case of the “tent-like” maps from [11]. In particular, if the key lemma shown in (4.1) can be replicated with an adapted potential function, then the arguments from the ordinary derivative situation can be mimicked to produce a proof of (5.2). DERIVATIVES OF SLIPPERY DEVIL’S STAIRCASES 365

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