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

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.)