The Hausdorff dimension of the nondifferentiability set of the Cantor function is [𝑙𝑛(2)/𝑙𝑛(3)]²

Abstract

Abstract.The main purpose of this note is to verify that the Hausdorff dimension of the set of points N* at which the Cantor function is not differentiable is [ln(2)/ln(3)]2 . It is also shown that the image of N* under the Cantor function has Hausdorff dimension ln(2)/ln(3).Similar results follow for a standard class of Cantor sets of positive measure and their corresponding Cantor functions.The Hausdorff dimension of the set of points N* at which the Cantor function is not differentiable is [ln(2)/ln(3)]2.Chapter 1 in [5] provides a nice introduction to Hausdorff measure and dimension; references [5][6][7] pursue the topic. We begin our proof with some notation and discussion. Let C denote the Cantor set. Let N+ (N~) denote the set of points at which the Cantor function does not have a right side (left side) derivative, finite or infinite. Then N* = N+ U N~ U {t: t is an end point of C} denotes the nondifferentiability set of the Cantor function. Although we will assume familiarity with [4], where Eidswick characterized N*, some material is repeated for completeness.A number t in C has a ternary representation t = (tx, ... , tt, ...), where /, = 0 or 2.Let z(az) denote the position of the Azth zero in the ternary representation of t;(la) If teN+,then limsup{z(AJ + 1)/z(aj)} > ln(3)/ln(2); (lb) If limsup{z(AJ + 1)/z(az)} > ln(3)/ln(2), then t e N+ . Let md denote the ^-dimensional Hausdorff measure, and put r = ln(2)/ln(3).We will compute the Hausdorff dimension of N* by verifying, then mdN* > Kd > 0; Kd will be specified later for a sequence of d 's increasing to r2. Condition (A) will be verified for each d satisfying the inequalities 1 > d > r2 by constructing a set E (depending on d) which contains N* and satisfies the equation mdE = 0. To verify (B), we will consider a sequence {d"} of

Hölder differentiability of self-conformal devil's staircases

“…[4]) where dim P is the packing measure and thus dim H K ≤γ(q) ≤ T (q) + qγ(q). As upper and lower bound coincide we have the required result (2). The case q < 0 is proven similarly and left to the reader.…”

Sets of nondifferentiability for conjugacies between expanding interval maps

“…(1) Φ is a C 1+ circle map; (2) dim H {ξ ∈ S 1 : Φ (ξ) does not exists in the generalised sense} = 0;…”

The infinite derivatives of Okamoto's self-affine functions: an application of $\beta$-expansions

“…For a ≥ 2/3, N (a) = (0, 1), so N (a) is actually ascending on [1/2, 1). The dimension of N (1/2) was computed by Darst [4]. That dim H N (a) = d(a) for a ∈ (1/2, a 0 ) follows from (25) and Lemma 4.2, noting that φ(a 0 ) = 1/3.…”

“…In most cases, formulas are given for the Hausdorff dimension of the sets ∆ ∞ (f ) and ∆ ∼ (f ) of points where the function f in question has, respectively, an infinite derivative, or neither a finite nor an infinite derivative. For the class of ordinary devil's staircases, this work was begun by Darst [4], who showed for f the classical Cantor function (F 1/2 in our notation) that dim H ∆ ∼ (f ) = (log 2/ log 3) 2 . Successive generalizations were obtained by Darst [5], Falconer [8], Kesseböhmer and Stratmann [14], and, most recently, Troscheit [28].…”