Weak separation condition, Assouad dimension, and Furstenberg homogeneity

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.

22 pages, 2 figures

References in corpus (3)

• Dimension and measures on sub-self-affine sets
• Separation conditions on controlled Moran constructions
• The Scenery Flow for Self-Affine Measures

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