A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra

Jonathan Fraser, Liam Stuart

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The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. We focus on the setting of geometrically finite Kleinian groups with parabolic elements and parabolic rational maps. In this context an especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two families.
Original languageEnglish
Pages (from-to)103-118
Number of pages16
JournalBulletin of the American Mathematical Society
Issue number1
Early online date2 Aug 2023
Publication statusPublished - 1 Jan 2024


  • Sullivan dictionary
  • Assouad dimension
  • Assouad spectrum
  • Kleinian group
  • Rational map
  • Julia set
  • Patterson-Sullivan measure
  • Conformal measure
  • Parabolicity


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