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≥2 affine maps on R2. These functions satisfy a functional equation of the form ϕ(akx+bk)=ckx+dkϕ(x)+ek, for k=1,2,…,r and x∈[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 ϕ is given by the multifractal formalism when |dk|≥|ak| for at least one k, but the multifractal formalism may fail otherwise, depending on the relationship between the shear parameters ck and the other parameters. In the special case when ak>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 (2018) [1] and Dubuc (2018) [6].