Hausdorff dimension and measures on Julia sets of some meromorphic maps

Krzysztof Barański

Fundamenta Mathematicae (1995)

  • Volume: 147, Issue: 3, page 239-260
  • ISSN: 0016-2736

Abstract

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We study the Julia sets for some periodic meromorphic maps, namely the maps of the form f ( z ) = h ( e x p 2 π i T z ) where h is a rational function or, equivalently, the maps ˜ f ( z ) = e x p ( 2 π i h ( z ) ) . When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1. From this we obtain some lower bound for HD(J(f)) and real-analyticity of HD(J(f)) with respect to f. As an example the family f λ ( z ) = λ t a n z is studied. We estimate H D ( J ( f λ ) ) near λ = 0 and show it is a monotone function of real λ.

How to cite

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Barański, Krzysztof. "Hausdorff dimension and measures on Julia sets of some meromorphic maps." Fundamenta Mathematicae 147.3 (1995): 239-260. <http://eudml.org/doc/212087>.

@article{Barański1995,
abstract = {We study the Julia sets for some periodic meromorphic maps, namely the maps of the form $f(z) = h(exp \frac\{2πi\}\{T\}z)$ where h is a rational function or, equivalently, the maps $˜f(z) = exp (\frac\{2πi\}\{h\}(z))$. When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1. From this we obtain some lower bound for HD(J(f)) and real-analyticity of HD(J(f)) with respect to f. As an example the family $f_λ(z)=λ tan z$ is studied. We estimate $HD(J(f_λ))$ near λ = 0 and show it is a monotone function of real λ.},
author = {Barański, Krzysztof},
journal = {Fundamenta Mathematicae},
keywords = {meromorphic maps; Julia sets; Hausdorff dimension},
language = {eng},
number = {3},
pages = {239-260},
title = {Hausdorff dimension and measures on Julia sets of some meromorphic maps},
url = {http://eudml.org/doc/212087},
volume = {147},
year = {1995},
}

TY - JOUR
AU - Barański, Krzysztof
TI - Hausdorff dimension and measures on Julia sets of some meromorphic maps
JO - Fundamenta Mathematicae
PY - 1995
VL - 147
IS - 3
SP - 239
EP - 260
AB - We study the Julia sets for some periodic meromorphic maps, namely the maps of the form $f(z) = h(exp \frac{2πi}{T}z)$ where h is a rational function or, equivalently, the maps $˜f(z) = exp (\frac{2πi}{h}(z))$. When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1. From this we obtain some lower bound for HD(J(f)) and real-analyticity of HD(J(f)) with respect to f. As an example the family $f_λ(z)=λ tan z$ is studied. We estimate $HD(J(f_λ))$ near λ = 0 and show it is a monotone function of real λ.
LA - eng
KW - meromorphic maps; Julia sets; Hausdorff dimension
UR - http://eudml.org/doc/212087
ER -

References

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  8. [MU] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, unpublished, 1994. 
  9. [Mc] C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), 329-342. Zbl0618.30027
  10. [PP] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990). 
  11. [P] F. Przytycki, On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions, Bol. Soc. Brasil. Mat. 20 (1990), 95-125. Zbl0723.58030
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