Hausdorff dimension and measures on Julia sets of some meromorphic maps

• Autorzy: Krzysztof Barański
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 28A78: Hausdorff and packing measures

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1995
• Tom: 147
• Numer: 3
• Strony: 239-260

Daty

wydano

1995

otrzymano

1994-06-14

Twórcy

autor

Krzysztof Barański

• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland,

Bibliografia

• [BKL] I. N. Baker, J. Kotus and Y. Lü, Iterates of meromorphic functions, I, Ergodic Theory Dynam. Systems 11 (1991), 241-248; II, J. London Math. Soc. (2) 42 (1990), 267-278; III, Ergodic Theory Dynam. Systems 11 (1991), 603-618.
• [B1] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, 1975.
• [B2] R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S. 50 (1979), 11-26.
• [DU] M. Denker and M. Urbański, Geometric measures for parabolic rational maps, Ergodic Theory Dynam. Systems 12 (1992), 53-66.
• [DK] R. L. Devaney and L. Keen, Dynamics of meromorphic maps: maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. (4) 22 (1989), 55-79.
• [G] G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Transl. Math. Monographs 26, Amer. Math. Soc., 1969.
• [K] J. Kotus, On the Hausdorff dimension of Julia sets of meromorphic functions, I, Bull. Soc. Math. France 122 (1994), 305-331; II, ibid. 123 (1995), 33-46.
• [MU] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, unpublished, 1994.
• [Mc] C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), 329-342.
• [PP] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990).
• [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.
• [R] D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), 99-107.