Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

Balázs Bárány, Károly Simon, Istvan Tamas Kolossvary, Michal Rams

Research output: Contribution to journalArticlepeer-review


This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.
Original languageEnglish
Pages (from-to)2051 - 2081
Number of pages31
JournalProceedings of the Royal Society of Edinburgh, Section A: Mathematics
Issue number6
Early online date15 Dec 2020
Publication statusPublished - Dec 2021


  • Self-conformal sets
  • Weak separation property
  • Assouad dimension


Dive into the research topics of 'Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line'. Together they form a unique fingerprint.

Cite this