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Abstract
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 language | English |
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Pages (from-to) | 103-118 |
Number of pages | 16 |
Journal | Bulletin of the American Mathematical Society |
Volume | 61 |
Issue number | 1 |
Early online date | 2 Aug 2023 |
DOIs | |
Publication status | Published - 1 Jan 2024 |
Keywords
- Sullivan dictionary
- Assouad dimension
- Assouad spectrum
- Kleinian group
- Rational map
- Julia set
- Patterson-Sullivan measure
- Conformal measure
- Parabolicity
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Dive into the research topics of 'A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra'. Together they form a unique fingerprint.Projects
- 2 Finished
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New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
1/09/19 → 31/01/23
Project: Standard
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Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
1/02/18 → 11/06/21
Project: Standard