Generic Hölder level sets and fractal conductivity

Abstract

Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the “thickness/narrow cross-sections” of a “network” corresponding to a fractal set, F. This lead to the definition of the topological Hausdorff dimension of fractals. In this paper we continue our study of the level sets of generic 1-Hölder-α functions. While in a previous paper we gave the initial definitions and established some properties of these generic level sets, in this paper we provide numerical estimates in the case of the Sierpiński triangle. These calculations give better insight and illustrate why can one think of these generic 1-Hölder-α level sets as something measuring “thickness/narrow cross-sections/conductivity” of a fractal “network”.We also give an example for the phenomenon which we call phase transition for D∗(α,F). This roughly means that for a certain lower range of αs only the geometry of F determines D∗(α,F) while for larger values the Hölder exponent, α also matters.