Assouad type dimensions of infinitely generated self-conformal sets

Amlan Banaji*, Jonathan Fraser

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of sets. Under natural separation conditions, we prove a formula for the Assouad dimension and prove sharp bounds for the Assouad spectrum in terms of the Hausdorff dimension of the limit set and dimensions of the set of fixed points of the contractions. The Assouad spectra of the family of examples which we use to show that the bounds are sharp display interesting behaviour, such as having two phase transitions. Our results apply in particular to sets of real or complex numbers which have continued fraction expansions with restricted entries, and to certain parabolic attractors.
Original languageEnglish
Article number045004
Number of pages32
Issue number4
Early online date29 Feb 2024
Publication statusPublished - Apr 2024


  • Conformal iterated function system
  • Assouad dimension
  • Assouad spectrum
  • Continued fractions
  • Parabolic iterated function system


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