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

TL;DR

This paper explores the differentiability properties of a two-parameter family of self-affine functions, revealing deep connections with noninteger base expansions and extending previous theorems to characterize the nature of their derivatives.

Contribution

It extends Okamoto's function to a two-parameter family and characterizes the sets where derivatives are zero or infinite, linking these to $eta$-expansions and Hausdorff dimensions.

Findings

The derivative of the functions is either 0, infinite, or undefined.

The set where the derivative is zero varies from empty to full measure depending on parameters.

The set of points with infinite derivative relates to unique $eta$-expansions and its Hausdorff dimension is computed.

Abstract

This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called {\em $\beta$-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 $\{F_{N,a}: N\in\mathbb{N}, a\in(0,1)\}$. We first show that for each $x$, $F_{N,a}'(x)$ is either $0$, $\pm\infty$, 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 $\{x: F_{N,a}'(x)=0\}$ 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 $\mathcal{D}_\infty(a):=\{x:F_{N,a}'(x)=\pm\infty\}$, which enables us to closely relate this set to the set of points which have a unique expansion in the (typically noninteger) base $\beta=1/a$. Recent advances in the theory of $\beta$-expansions are then used to determine the cardinality and Hausdorff dimension of $\mathcal{D}_\infty(a)$, which depends qualitatively on the value of $a$ relative to a second pair of thresholds.