A dichotomy of self-conformal subsets of the real line with overlaps

TL;DR

This paper establishes a dichotomy for self-conformal subsets of the real line, showing that those lacking the weak separation condition have maximal Assouad dimension, linking geometric properties with separation conditions.

Contribution

It proves that non-weakly separated self-conformal sets have full Assouad dimension and characterizes when Hausdorff and Assouad dimensions coincide for such sets.

Findings

Self-conformal sets without weak separation have full Assouad dimension.

A dichotomy exists: for these sets, either Hausdorff and Assouad dimensions agree or Assouad dimension is 1.

Weak separation property is equivalent to equality of Hausdorff and Assouad dimensions in this context.

Abstract

We show that self-conformal subsets of $\mathbb{R}$ that do not satisfy the weak separation condition have full Assouad dimension. Combining this with a recent results by K\"aenm\"aki and Rossi we conclude that an interesting dichotomy applies to self-conformal and not just self-similar sets: if $F\subset\mathbb{R}$ is self-conformal with Hausdorff dimension strictly less than $1$, either the Hausdorff dimension and Assouad dimension agree or the Assouad dimension is $1$. We conclude that the weak separation property is in this case equivalent to Assouad and Hausdorff dimension coinciding. (This manuscript contains errors, see comment below.)