The pointwise Hölder spectrum of general self-affine functions on an interval

Abstract

This paper gives the pointwise Hölder (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\ge 2$ affine maps on ${\mathbb{R}}^2$. These functions satisfy a functional equation of the form $\varphi (a_{k}x+b_k)=c_{k}x+d_k\varphi (x)+e_k$, 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 $\varphi$ is given by the multifractal formalism when $|d_k|\ge |a_k|$ 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 Hölder 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].

40 pages, 3 figures. The Introduction has been reorganized somewhat