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Measures and dimensions in conformal dynamics

Author(s): Mariusz Urbanski
Journal: Bull. Amer. Math. Soc. 40 (2003), 281-321.
MSC (2000): Primary 35F35, 37D35; Secondary 37F15, 37D20, 37D25, 37D45, 37A40, 37A05
Posted: April 8, 2003
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Abstract: This survey collects basic results concerning fractal and ergodic properties of Julia sets of rational functions of the Riemann sphere. Frequently these results are compared with their counterparts in the theory of Kleinian groups, and this enlarges the famous Sullivan dictionary. The topics concerning Hausdorff and packing measures and dimensions are given most attention. Then, conformal measures are constructed and their relations with Hausdorff and packing measures are discussed throughout the entire article. Also invariant measures absolutely continuous with respect to conformal measures are touched on. While the survey begins with facts concerning all rational functions, much time is devoted toward presenting the well-developed theory of hyperbolic and parabolic maps, and in Section 3 the class NCP is dealt with. This class consists of such rational functions $f$ that all critical points of $f$ which are contained in the Julia set of $f$ are non-recurrent. The NCP class comprises in particular hyperbolic, parabolic and subhyperbolic maps. Our last section collects some recent results about other subclasses of rational functions, e.g. Collet-Eckmann maps and Fibonacci maps. At the end of this article two appendices are included which are only loosely related to Sections 1-4. They contain a short description of tame mappings and the theory of equilibrium states and Perron-Frobenius operators associated with Hölder continuous potentials.

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Additional Information:

Mariusz Urbanski
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430
Email: urbanski@unt.edu

DOI: 10.1090/S0273-0979-03-00985-6
PII: S 0273-0979(03)00985-6
Received by editor(s): December 22, 1999,
Received by editor(s) in revised form: January 8, 2003
Posted: April 8, 2003
Additional Notes: Research partially supported by NSF Grant DMS 9801583
Copyright of article: Copyright 2003, American Mathematical Society

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