Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 465-490

WEAK SEPARATION CONDITION, ASSOUAD DIMENSION, AND FURSTENBERG HOMOGENEITY

Antti Käenmäki and Eino Rossi

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; antti.kaenmaki 'at' jyu.fi

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; eino.rossi 'at' jyu.fi

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.

2010 Mathematics Subject Classification: Primary 28A80; Secondary 37C45, 28D05, 28A50.

Key words: Moran construction, iterated function system, weak separation condition, dimension.

Reference to this article: A. Käenmäki and E. Rossi: Weak separation condition, Assouad dimension, and Furstenberg homogeneity. Ann. Acad. Sci. Fenn. Math. 41 (2016), 465-490.

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doi:10.5186/aasfm.2016.4133