Measures and dimensions in conformal dynamics

By Mariusz Urba'nski. 2003

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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. 1. Dimensions of Julia sets A first issue we will be dealing with in this article is to describe various fractals of Julia sets as captured by the Hausdorff, packing and box dimensions. Afterwards, we address the question of when the corresponding Hausdorff and packing measures are positive and finite. Given a subset A of a metric space (X, d) a countable family {B(xi, ri)}i=1 of open balls centered at points of A is said to be a packing of A if and only if for any pair i 6= j d(xi, xj) ≥ ri + rj . The supremum sup{ri : i ≥ 1} is called the radius of the packing {B(xi, ri)}i=1. Given in addition a positive radius r > 0, denote by N(A, r) the minimal number of open balls with radius r needed to cover A and by P (A, r) the maximal number of open balls with radius r forming a packing of A. In order to get an idea of what the box dimension is, imagine a two-dimensional smooth surface A in the Euclidean space R. It is reasonable to expect the minimal number N(A, r) and the maximal number P (A, r) to be some multiples of r−2. The Received by the editors December 22, 1999, and, in revised form, January 8, 2003. 2000 Mathematics Subject Classification. Primary 35F35, 37D35; Secondary 37F15, 37D20, 37D25, 37D45, 37A40, 37A05. Research partially supported by NSF Grant DMS 9801583. c ©2003 American Mathematical Society 281 282 MARIUSZ URBAŃSKI coefficient is not important to understand dimensionality of A, but the exponent is crucial, and it is captured by the following limits: lim r→0 logN(A, r) − log r = lim r→0 logP (A, r) − log r . In general these limits fail to exist, and the following two quantities are defined: BD(A) = lim inf r→0 logN(A, r) − log r = lim inf r→0 logP (A, r) − log r

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9ccd01ba6b9f94c7fc8913d6cad62e56265a929a

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In Mathematical Physics

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In Statistical and Nonlinear Physics

  1. K. Gelfert
  2. Luís Barreira
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  4. Daniel Thompson
  5. B. Saussol
  6. Daniel J. Thompson
  7. Wen Huang

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