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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Martin boundaries of Denjoy domains


Author: Shigeo Segawa
Journal: Proc. Amer. Math. Soc. 103 (1988), 177-183
MSC: Primary 31C35
DOI: https://doi.org/10.1090/S0002-9939-1988-0938665-2
MathSciNet review: 938665
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Abstract: Let $ E( \subset {\mathbf{\hat C}})$ be a compact set in the real axis. It is shown that there exists an $ E$ with zero linear measure such that Martin compactification of the domain $ {\mathbf{\hat C}} - E$ is not homeomorphic to $ {\mathbf{\hat C}}$. Moreover, it is shown that if for some $ \lambda > \tfrac{1}{2}$

$\displaystyle \frac{{\vert{E^c} \cap [ - t,t]\vert}}{t} = O\left( {{{\left( {\frac{1}{{\log {t^{ - 1}}}}} \right)}^\lambda }} \right)\quad (t \to 0),$

the set of minimal Martin boundary points of $ {\mathbf{\hat C}} - E$ 'over 0' consists of two points. This assertion is not valid for $ \lambda = \tfrac{1}{2}$.

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DOI: https://doi.org/10.1090/S0002-9939-1988-0938665-2
Article copyright: © Copyright 1988 American Mathematical Society