Fractal dimensions of graph of Weierstrass-type function and local H\"older exponent spectra

TL;DR

This paper investigates the fractal dimensions and local H"older exponent spectra of Weierstrass-type functions, providing formulas for their box and Hausdorff dimensions, including in randomized cases, using thermodynamic formalism.

Contribution

It introduces new formulas for fractal dimensions and spectra of Weierstrass-type functions, including in randomised settings, expanding understanding of their fractal properties.

Findings

Determined box dimension of the graph of W

Characterized Hausdorff spectrum of local H"older exponents

Provided a novel formula for the lifted Hausdorff spectrum in randomized cases

Abstract

We study several fractal properties of the Weierstrass-type function \[ W(x)=\sum_{n=0} ^\infty \lambda (x) \lambda(\tau x) \cdots \lambda (\tau ^{n-1}x)\, g(\tau ^n x), \] where $\tau :[0,1)\to[0,1)$ is a cookie cutter map with possibly fractal repeller, and $\lambda$ and $g$ are functions with proper regularity. In the first part, we determine the box dimension of the graph of $W$ and Hausdorff dimension of its randomised version. In the second part, the Housdorff spectrum of the local H\"older exponent is characterised in terms of thermodynamic formalisms. Furthermore, in the randomised case, a novel formula for the lifted Hausdorff spectrum on the graph is provided.