We consider the compact plane sets known as Swiss cheese sets, which are a useful source of examples in the theory of uniform algebras and rational approximation. We develop a theory of allocation maps connected to such sets and we use this theory to modify examples previously constructed in the literature to obtain examples homeomorphic to the Sierpiński carpet. Our techniques also allow us to avoid certain technical difficulties in the literature.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-3-5,
author = {J. F. Feinstein and M. J. Heath},
title = {Swiss cheeses, rational approximation and universal plane curves},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {289-306},
zbl = {1202.46056},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-3-5}
}
J. F. Feinstein; M. J. Heath. Swiss cheeses, rational approximation and universal plane curves. Studia Mathematica, Tome 196 (2010) pp. 289-306. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-3-5/