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 Käenmäki 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.)

The proofs as well as the statement of some of the theorems in this manuscript contain errors. For a corrected version, see arXiv:1803.09113v2

References in corpus (1)

• Weak separation condition, Assouad dimension, and Furstenberg homogeneity