Harmonic-measure distribution functions for a class of multiply connected symmetrical slit domains
Green, Christopher C. Snipes, Marie A. Ward, Lesley A. Crowdy, Darren G.
Green, Christopher C.
Snipes, Marie A.
Ward, Lesley A.
Crowdy, Darren G.
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2022-03-02
Type
Article
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Keywords
Conformal map,Prime function,Multiply connected slit domain,Harmonic-measure distribution function
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Citation
Green Christopher C., Snipes Marie A., Ward Lesley A. and Crowdy Darren G. 2022Harmonic-measure distribution functions for a class of multiply connected symmetrical slit domains Proc. R. Soc. A.4782021083220210832 http://doi.org/10.1098/rspa.2021.0832
Abstract
The harmonic-measure distribution function, or h-function, of a planar domain Ω ⊂ C with respect to a basepoint z0 ∈ Ω is
a signature that profiles the behaviour in Ω of a Brownian particle starting from z0. Explicit calculation of h-functions for a
wide array of simply connected domains using conformal mapping techniques has allowed many rich connections to be made
between the geometry of the domain and the behaviour of its h-function. Until now, almost all h-function computations
have been confined to simply connected domains. In this work, we apply the theory of the Schottky-Klein prime function to
explicitly compute the h-function of the doubly connected slit domain C \ ([−1/2, −1/6] ∪ [1/6, 1/2]). In view of the
connection between the middle-thirds Cantor set and highly multiply connected symmetric slit domains, we then extend
our methodology to explicitly construct the h-functions associated with symmetric slit domains of arbitrary even
connectivity. To highlight both the versatility and generality of our results, we graph the h-functions associated with
quadruply and octuply connected slit domains.
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Publisher
Royal Society
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Series
Proceedings of the Royal Society;Volume 478 Issue 2259
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1364-5021