Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

TL;DR

This paper studies the geometric properties of generic self-conformal IFSs on the real line, showing that typically their attractors with Hausdorff dimension less than one have zero measure and maximal Assouad dimension, especially when cylinders intersect.

Contribution

It demonstrates that generically, intersecting cylinders lead to failure of the weak separation property, enabling the application of recent theoretical results to describe the attractor's dimensions.

Findings

Attractors with Hausdorff dimension less than 1 have zero Hausdorff measure.

Generically, the Assouad dimension of such attractors is equal to 1.

The phenomenon holds for transversal families, including translation families, in both topological and measure-theoretic senses.

Abstract

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than $1$ then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to $1$. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, K\"aenm\"aki and Troscheit [BLMS, 2020]. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.