6533b858fe1ef96bd12b647f

RESEARCH PRODUCT

Weak separation condition, Assouad dimension, and Furstenberg homogeneity

Eino RossiAntti Käenmäki

subject

General MathematicsHomogeneity (statistics)ta111Open setPrimary 28A80 Secondary 37C45 28D05 28A50Moran constructioniterated function systemSet (abstract data type)CombinatoricsDimension (vector space)dimensionMathematics - Classical Analysis and ODEsweak separation conditionClassical Analysis and ODEs (math.CA)FOS: MathematicsLimit (mathematics)Limit setCluster analysisReal lineMathematics

description

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.

yearjournalcountryeditionlanguage
2015-06-25
10.5186/aasfm.2016.4133http://arxiv.org/abs/1506.07851