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arXiv · 1602.05821

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

Jasmina Angelevska·Sascha Troscheit
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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.)

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Jasmina Angelevska, Sascha Troscheit. 2019-05-01. A dichotomy of self-conformal subsets of the real line with overlaps. https://arxiv.org/abs/1602.05821

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