Generic H\"older level sets and fractal conductivity

TL;DR

This paper investigates the Hausdorff dimensions of level sets of generic 1-H"older-$eta$ functions on fractals, providing numerical estimates for the Sierpiński triangle and exploring phase transitions in fractal conductivity related to these dimensions.

Contribution

It extends previous work by providing numerical estimates for level set dimensions on the Sierpiński triangle and introduces the concept of phase transition in the fractal dimension depending on the H"older exponent.

Findings

Numerical estimates for level set dimensions on the Sierpiński triangle.

Identification of phase transition phenomena in fractal dimensions.

Insight into how H"older regularity influences 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 $1$-H\"older-$\alpha$ 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 Sierpi\'nski triangle. These calculations give better insight and illustrate why can one think of these generic $1$-H\"older-$\alpha$ 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 $D_{*}(\alpha, F)$. This roughly means that for a certain lower range { of $\alpha$s } only the geometry of $F$ determines $D_{*}(\alpha, F)$ while for larger values the H\"older exponent, $\alpha$ also matters.