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Geometric measures for parabolic rational maps

  • M. Denker (a1) and M. Urbański (a2)

Let h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.

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[2]P. Blanchard . Complex analytic dynamics of the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85141.

[3]H. Brolin . Invariant sets under iteration of rational functions. Ark. Mat. 6 (1965), 103144.

[5]M. Denker & M. Urbański . Absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points. Forum Math. 3 (1991), 561579.

[6]M. Guzmán . Differentiation of integrals in ℝn. Springer Lecture Notes in Mathematics 481, Springer Verlag, 1975.

[9]D. Sullivan . Conformal dynamical systems. In: Geometric Dynamics. Springer Lecture Notes in Mathematics1007, Springer Verlag, 1983, pp 725752.

[10]D. Sullivan . Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259277.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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