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arXiv · 1506.07851

Weak separation condition, Assouad dimension, and Furstenberg homogeneity

Antti Käenmäki·Eino Rossi
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Abstract

We consider dimensional properties of limit sets of Moran constructions satisfying the finite clustering property. Just to name a few, such limit sets include self-conformal sets satisfying the weak separation condition and certain sub-self-affine sets. In addition to dimension results for the limit set, we manage to express the Assouad dimension of any closed subset of a self-conformal set by means of the Hausdorff dimension. As an interesting consequence of this, we show that a Furstenberg homogeneous self-similar set in the real line satisfies the weak separation condition. We also exhibit a self-similar set which satisfies the open set condition but fails to be Furstenberg homogeneous.

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BibTeXRIS

Antti Käenmäki, Eino Rossi. 2015-06-25. Weak separation condition, Assouad dimension, and Furstenberg homogeneity. https://doi.org/10.5186/aasfm.2016.4133

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