The geometry of conformal measures for parabolic rational maps

Bernd O Stratmann, M Urbánski

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

We study the h-conformal measure for parabolic rational maps, where h denotes the Hausdorff dimension of the associated Julia sets. We derive a formula which describes in a uniform way the scaling of this measure at arbitrary elements of the Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational maps (the analogue of the 'logarithmic law for geodesics' in the theory of Kleinian groups) and show that this law provides some efficient control for the fluctuation of the h-conformal measure. We then show that these results lead to some refinements of the description of this measure in terms of Hausdorff and packing measures with respect to some gauge functions. Also, we derive a simple proof of the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the conformal measure is a regular doubling measure, we show that its Renyi dimension and its information dimension are equal to h and we compute its logarithmic index.

Original languageEnglish
Pages (from-to)141-156
Number of pages16
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume128
DOIs
Publication statusPublished - Jan 2000

Keywords

  • KLEINIAN-GROUPS
  • JULIA SETS
  • HAUSDORFF
  • POINTS
  • OLD

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