The pointwise H\"older spectrum of general self-affine functions on an interval

Abstract

This paper gives the pointwise H\"older (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These functions satisfy a functional equation of the form $\phi(a_k x+b_k)=c_k x+d_k\phi(x)+e_k$, for $k=1,2,\dots,r$ and $x\in[0,1]$. They include the Takagi function, the Riesz-Nagy singular functions, Okamoto's functions, and many other well-known examples. It is shown that the multifractal spectrum of $\phi$ is given by the multifractal formalism when $|d_k|\geq |a_k|$ for at least one $k$, but the multifractal formalism may fail otherwise, depending on the relationship between the shear parameters $c_k$ and the other parameters. In the special case when $a_k>0$ for every $k$, an exact expression is derived for the pointwise H\"older exponent at any point. These results extend recent work by the author [Adv. Math. 328 (2018), 1-39] and S. Dubuc [Expo. Math. 36 (2018), 119-142].