Epsilon-Distortion Complexity for Cantor Sets

TL;DR

This paper introduces epsilon-distortion complexity to quantify how concisely Cantor sets can be described and estimates this complexity for various types of Cantor sets generated by iterated function systems, revealing dependencies on smoothness and dimension.

Contribution

It defines epsilon-distortion complexity for sets and provides estimates for different Cantor sets, highlighting dependencies on smoothness and fractal dimension.

Findings

Complexity depends on the smoothness class of the Cantor set.

Complexity varies with the box counting dimension.

Polynomial and random affine Cantor sets exhibit different complexity behaviors.

Abstract

We define the epsilon-distortion complexity of a set as the shortest program, running on a universal Turing machine, which produces this set at the precision epsilon in the sense of Hausdorff distance. Then, we estimate the epsilon-distortion complexity of various central Cantor sets on the line generated by iterated function systems (IFS's). In particular, the epsilon-distortion complexity of a C^k Cantor set depends, in general, on k and on its box counting dimension, contrarily to Cantor sets generated by polynomial IFS or random affine Cantor sets.