Hölder Spectrum of Typical Monotone Continuous Functions

Abstract

We compute the Hölder spectrum and the moment sums of the typical (in the sense of Baire category) monotone continuous function, $f$. We show that these functions are of multifractal nature, for example $\alpha$ equals the Hausdorff dimension of those points where such a function, $f$ is of exact Hölder class $\alpha\in [0,1].$ We also prove that $f$ "grows" only at those points where it has bad Hölder properties, that is, at those points where it is of Hölder class $0.$ The upper moment sum $\overline{\tau}_f (q)=0$ if $q <0,$ $\overline{\tau}_f (q)=0=1-q$ if $0\leq q\leq 1,$ and $\overline{\tau}_f (q)=0$ if $1< q.$ The lower moment sum $\underline{\tau}_f (q)=1-q$ if $q<0,$ $\underline{\tau}_f (q)=0$ if $0\leq q\leq 1,$ and $\underline{\tau}_f (q)=1-q$ if $1<q.$