Manfred Denker, R. Mauldin, Z. Nitecki, Mariusz Urbański (1998)
Fundamenta Mathematicae
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We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.
Henk Bruin, Weixiao Shen, Sebastian Van Strien (2006)
Annales scientifiques de l'École Normale Supérieure
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M. Urbański (1990)
Studia Mathematica
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Mary Rees (1986)
Annales scientifiques de l'École Normale Supérieure
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Lei Tan (2002)
Annales scientifiques de l'École Normale Supérieure
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Huaibin Li, Weixiao Shen (2008)
Fundamenta Mathematicae
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Consider a rational map f on the Riemann sphere of degree at least 2 which has no parabolic periodic points. Assuming that f has Rivera-Letelier's backward contraction property with an arbitrarily large constant, we show that the upper box dimension of the Julia set J(f) is equal to its hyperbolic dimension, by investigating the properties of conformal measures on the Julia set.
Dennis Sullivan (1979)
Publications Mathématiques de l'IHÉS
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Zoltán M. Balogh, Matthieu Rickly, Francesco Serra Cassano (2003)
Publicacions Matemàtiques
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We compare the Hausdorff measures and dimensions with respect to the Euclidean and Heisenberg metrics on the first Heisenberg group. The result is a dimension jump described by two inequalities. The sharpness of our estimates is shown by examples. Moreover a comparison between Euclidean and H-rectifiability is given.