Harmonic-measure distribution functions for a class of multiply connected symmetrical slit domains
Date
2022-03-02Author
Green, Christopher C.
Snipes, Marie A.
Ward, Lesley A.
Crowdy, Darren G.
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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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