Dimensions of Julia sets of meromorphic functions with finitely many poles. (English) Zbl 1087.37046
Summary: Let \(f\) be a transcendental meromorphic function with finitely many poles such that the finite singularities of \(f^{-1}\) lie in a bounded set. We show that the Julia set of \(f\) has Hausdorff dimension strictly greater than one and packing dimension equal to two. The proof for Hausdorff dimension simplifies the earlier argument given for transcendental entire functions [the second author, Math. Proc. Camb. Philos. Soc. 119, 513–536 (1996; Zbl 0852.30018)],
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |
| 37F35 | Conformal densities and Hausdorff dimension for holomorphic dynamical systems |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
