Differentiability of a two-parameter family of self-affine functions

Abstract

This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called β-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 {FN,a:N∈N,1/(N+1)<a<1}. We first show that for each x, FN,a′(x) is either 0, ±∞, or undefined. We then extend Okamoto's theorem by proving that for each N, depending on the value of a relative to a pair of thresholds, the set {x:FN,a′(x)=0} 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 D∞(a):={x:FN,a′(x)=±∞}, which enables us to closely relate this set to the set of points which have a unique expansion in the (typically noninteger) base β=1/a. Recent advances in the theory of β-expansions are then used to determine the cardinality and Hausdorff dimension of D∞(a), which depend qualitatively on the value of a relative to a second pair of thresholds.