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arXiv · 1701.08571

On the geometric structure of the limit set of conformal iterated function systems

Antti Käenmäki
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Abstract

We consider infinite conformal iterated function systems on $\mathbb{R}^d$. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some $l$-dimensional $C^1$-submanifold with positive Hausdorff $t$-dimensional measure, where $0<l<d$ and $t$ is the Hausdorff dimension of the limit set. We then show that the closure of the limit set belongs to some $l$-dimensional affine subspace or geometric sphere whenever $d$ exceeds $2$ and analytic curve if $d$ equals $2$.

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Antti Käenmäki. 2017-01-30. On the geometric structure of the limit set of conformal iterated function systems. https://arxiv.org/abs/1701.08571

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