The infinite partition of a line segment and multifractal objects

TL;DR

This paper introduces an algorithm for partitioning a line segment based on a ratio, resulting in a multifractal object with a calculable spectrum, revealing detailed fractal properties in the infinite limit.

Contribution

The paper presents a novel algorithm for segment partitioning that produces multifractal structures and derives their fractal spectrum analytically.

Findings

The partitioning algorithm produces a binomial distribution of segment lengths.

The multifractal spectrum $D_k$ is analytically derived and its maximum identified.

Explicit values of $D_k$ are calculated for the limits $k/n o 0$ and 1.

Abstract

We report an algorithm for the partition of a line segment according to a given ratio $\nu$. At each step the length distribution among sets of the partition follows a binomial distribution. We call $k$-set to the set of elements with the same length at the step $n$. The total number of elements is $2^n$ and the number of elements in a same $k$-set is $C_n^k$. In the limit of an infinite partion this object become a multifractal where each $k$-set originate a fractal. We find the fractal spectrum $D_k$ and calculate where is its maximum. Finally we find the values of $D_k$ for the limits $k/n \to 0$ and 1.