Special families of piecewise linear iterated function systems

TL;DR

This paper studies the dimension and measure properties of specific families of piecewise linear iterated function systems, revealing conditions under which their attractors have positive Lebesgue measure and explicit Hausdorff dimension.

Contribution

It establishes a link between the Hausdorff dimension and exponential growth rate for certain systems, and shows positivity of Lebesgue measure for typical parameters.

Findings

Hausdorff dimension equals exponential growth rate for one family

Lebesgue measure is positive for typical parameters when dimension exceeds 1

Results apply to systems with positive contraction ratios

Abstract

This paper investigates the dimension theory of some families of continuous piecewise linear iterated function systems. For one family, we show that the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained from the most natural covering system. We also prove that for Lebesgue typical parameters, the 1-dimensional Lebesgue measure of the underlying attractor is positive, if this number is bigger than 1 and all the contraction ratios are positive.