The Hausdorff dimension of the nondifferentiability set of a non-symmetric Cantor function

Abstract

Each choice of numbers a and c in the segment (0, 1/2) produces a Cantor set Cac by recursively removing segments from the interior of the interval [0,1] so that intervals of relative length a and c remain on the left and right sides of the removed segment, respectively. A Cantor function Φac is obtained from Cac in much the same way that the standard Cantor function. Φ. is obtained from the Cantor ternary set. When a = c = 1/3. Cac is the Cantor ternary set. C, and Φuc is the standard Cantor function, Φ. The derivative of Φ is zero off C. and the upper derivative is infinite on C: the set N = {x ∈ C | the lower derivative of Φ is finite} has Hausdorff dimension [In 2/ In3]2. In this paper, similar results are established for .V*. the nondifferentiability set of Φac. The Hausdorff dimension of .V* is the maximum of the real numbers satisfying the following equations: r(In(1/c))2 = In((a + c)/c) In((a/c)x + 1). and x(In(1/a))2' = In((a + c)/a) In((c/a)x + 1).