Dimensions of Julia sets of meromorphic functions

Rippon, P.J. and Stallard, G.M. (2005). Dimensions of Julia sets of meromorphic functions. Journal of the London Mathematical Society, 71(3) pp. 669–683.

DOI: https://doi.org/10.1112/S0024610705006393

Abstract

We show that for any meromorphic function $f$ the Julia set $J(f)$ has constant local upper and lower box dimensions, $\overline{d}(J(f))$ and $\underline{d}(J(f))$, respectively, near all points of $J(f)$ with at most two
exceptions. Further, the packing dimension of the Julia set is equal to $\overline{d}(J(f))$. Using this result we show that, for any transcendental entire function $f$ in the class $B$ (that is, the class of functions such that the singularities of the inverse function are bounded), both the local upper box dimension and packing dimension of $J(f)$ are equal to 2. Our approach is to show that the subset of the Julia set containing those points that escape to infinity as quickly as possible has local upper box dimension equal to 2.

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