Generic H\"older level sets on Fractals

TL;DR

This paper investigates the Hausdorff dimensions of level sets of generic 1-Hölder-α functions on fractals, introducing a new measure D_*(α, F) to quantify the typical level set dimension and exploring its properties.

Contribution

It defines and analyzes the new concept D_*(α, F), relating it to the Hausdorff dimension of level sets for generic Hölder functions on fractals, including explicit calculations and properties.

Findings

D_*(α, F) measures the typical Hausdorff dimension of level sets.

For connected self-similar fractals, D_*(α, F) equals the Hausdorff dimension of almost all level sets.

The paper provides calculations for specific fractals like thick fractal sponges.

Abstract

Hausdorff dimensions of level sets of generic continuous functions defined on fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes. In those papers the topological Hausdorff dimension of fractals was defined. In this paper we start to study level sets of generic $1$-H\"older-$\alpha$ functions defined on fractals. This is related to some sort of "thickness", "conductivity" properties of fractals. The main concept of our paper is $D_{*}(\alpha, F)$ which is the essential supremum of the Hausdorff dimensions of the level sets of a generic $1$-H\"older-$\alpha$ function defined on the fractal $F$. We prove some basic properties of $D_{*}(\alpha, F)$, we calculate its value for an example of a "thick fractal sponge", we show that for connected self similar sets $D_{*}(\alpha, F)$ it equals the Hausdorff dimension of almost every level in the range of a generic $1$-H\"older-$\alpha$ function.