Piecewise linear iterated function systems on the line of overlapping construction

TL;DR

This paper investigates the Hausdorff dimension of attractors generated by piecewise linear iterated function systems on the real line, showing that under certain conditions, the dimension matches a natural exponent for most parameter choices.

Contribution

It establishes that for a broad class of overlapping piecewise linear IFS, the Hausdorff dimension equals a natural exponent for almost all parameters, extending understanding beyond separation conditions.

Findings

Hausdorff dimension equals the natural exponent for most parameters

Dimension result holds without separation conditions

Exceptional set of parameters has less than full packing dimension

Abstract

In this paper we consider Iterated Function Systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.