Fractal dimensions of continuous piecewise linear iterated function systems

TL;DR

This paper investigates the fractal dimensions of attractors generated by continuous piecewise linear iterated function systems on the real line, establishing conditions under which Hausdorff and box dimensions coincide.

Contribution

It provides a new result linking the Hausdorff and box dimensions of such attractors to a natural covering exponent under mild separation conditions.

Findings

Hausdorff and box dimensions are equal under the given conditions

Dimensions are given by the minimum of 1 and a specific exponent

The results apply to a broad class of piecewise linear systems

Abstract

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.