Intermediate dimensions of Moran sets and their visualization

TL;DR

This paper investigates the intermediate fractal dimensions of Moran sets, providing formulas for upper and lower dimensions, and visualizes their upper intermediate dimensions using Mobius transformations.

Contribution

It introduces a method to calculate upper and lower intermediate dimensions of Moran sets and visualizes these dimensions for homogeneous Moran sets.

Findings

Derived simplified formulas for intermediate dimensions of homogeneous Moran sets.

Established that upper intermediate dimensions can be expressed via Mobius transformations.

Showed the existence and calculation of upper and lower intermediate dimensions for Moran sets.

Abstract

Intermediate dimensions are a class of new fractal dimensions which provide a spectrum of dimensions interpolating between the Hausdorff and box-counting dimensions. In this paper, we study the intermediate dimensions of Moran sets. Moran sets may be regarded as a generalization of self-similar sets generated by using different class of similar mappings at each level with unfixed translations, and this causes the lack of ergodic properties on Moran set. Therefore, the intermediate dimensions do not necessarily exist, and we calculate the upper and lower intermediate dimensions of Moran sets. In particular, we obtain a simplified intermediate dimension formula for homogeneous Moran sets. Moreover, we study the visualization of the upper intermediate dimensions for some homogeneous Moran sets, and we show that their upper intermediate dimensions are given by Mobius transformations.