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The dichotomy of Hausdorff measures and equilibrium states for parabolic rational maps

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Book cover Ergodic Theory and Related Topics III

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1514))

Abstract

Let \(T:\bar {\mathbb{C}} \to \bar {\mathbb{C}}\)be a rational parabolic map of the Riemann sphere \(\bar {\mathbb{C}}\)and let \(f:\bar {\mathbb{C}} \to {\mathbb{R}}\)be a Lipschitz continuous function satisfying \(P(T,f) > \sup _{x \in \bar {\mathbb{C}}} f(x)\), where P(T, f) denotes the topological pressure. We show that the only equilibrium state for T and f is singular with respect to Hausdorff measures defined by functions of the upper class, and it is absolutely continuous with respect to Hausdorff measures defined by functions of the lower class.

Research supported by SFB 170, University of Göttingen

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Authors and Affiliations

  1. Institut für Mathematische Stochastik, University of Göttingen, Lotzestr. 13, W-3400, Göttingen, Germany

    Manfred Denker

  2. Instytut Matematyki, University of Toruń, Ul. Chopina 12/18, 87-100, Toruń, Poland

    Mariusz Urbański

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  1. Manfred Denker
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  2. Mariusz Urbański
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Ulrich Krengel Karin Richter Volker Warstat

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© 1992 Springer-Verlag

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Denker, M., Urbański, M. (1992). The dichotomy of Hausdorff measures and equilibrium states for parabolic rational maps. In: Krengel, U., Richter, K., Warstat, V. (eds) Ergodic Theory and Related Topics III. Lecture Notes in Mathematics, vol 1514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097530

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  • DOI: https://doi.org/10.1007/BFb0097530

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  • Print ISBN: 978-3-540-55444-8

  • Online ISBN: 978-3-540-47076-2

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