WEAK SEPARATION CONDITION, ASSOUAD DIMENSION, AND FURSTENBERG HOMOGENEITY

By A. Kaenmaki, Eino Rossi. 2015

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We consider dimensional properties of limit sets of Moran constructions satisfy- ing 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-ane 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 Hausdor 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/010ced037aa51e720dfa349a535de6edeb3abcac