We give a complete description of the possible Hausdorff dimensions of escaping sets for meromorphic functions with a finite number of singular values. More precisely, for any given
d
∈
[
0
,
2
]
d\in [0,2]
we show that there exists such a meromorphic function for which the Hausdorff dimension of the escaping set is equal to
d
d
. The main ingredient is to glue together suitable meromorphic functions by using quasiconformal mappings. Moreover, we show that there are uncountably many quasiconformally equivalent meromorphic functions for which the escaping sets have different Hausdorff dimensions.
A function which is transcendental and meromorphic in the plane has at least two singular values. On the one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either
$2$
or
$1/2$
. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in
$[0,2]$
(cf. [M. Aspenberg and W. Cui. Hausdorff dimension of escaping sets of meromorphic functions. Trans. Amer. Math. Soc.374(9) (2021), 6145–6178]). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than
$4$
.
We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second order differential equations in the complex plane.Mathematics Subject Classification: 30D05 (primary), 37F10 (secondary).
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