A Weak Local Irregularity Property in Sν Spaces

It has been shown that, from the prevalence point of view, elements of the Sν spaces are almost surely multifractal, while the Hölder exponent at almost every point is almost surely equal to the maximum Hölder exponent. We show here that typical elements of Sν are very irregular by proving that they almost surely satisfy a weak irregularity property: there exists a local irregularity exponent which is constant for almost every element of Sν and equal to the lowest Hölder exponent