Extending Real Maps Defined on a Subset of a Disk

Ball, B. J.; Ford, Jo; Thomas, E. S.

doi:10.2307/2035965

Proceedings of the American Mathematical Society

1969

DOI: 10.2307/2035965

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Extending Real Maps Defined on a Subset of a Disk

B. J. Ball¹,

Jo Ford²,

E. S. Thomas³

Abstract: A(x, y) = (x2+y2)112 and 7r(x, y) =x. Suppose Fis a subset of a disk D and / is a mapping of T into [O, l]; we are concerned here with conditions under which / can be extended to a mapping of D onto [0, l] which is topologically equivalent to A or to it. Recall that a mapping u: A->B is topologically equivalent to a mapping w: A'-tB' if there exist homeomorphisms h: A->A' and k: B-*B' such that

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Aligning functions defined on Cantor sets

Ford¹,

Thomas²

1969

Trans. Amer. Math. Soc.

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The «-sphere, Sn, is considered as the boundary of 7n + 1. The combinatorial boundary and interior of a cell B are denoted by dB and int B respectively. If F is an w-cell lying in 7n then the interior of F relative to In will be denoted : int B, rel 7", or simply by: intF. Frequently we shall consider 7n as the product P~1xl. Ifp e I, then, to conserve space, we shall write In~1xp rather than In~1x{p}. The symbol 77 denotes the natural projection of In onto 7""1 xO. Let/: K-^-I11'1 be a function whose domain K lies in In. We say that/can be aligned provided there is a self-homeomorphism u of 7n such that nu/K^f, where 7" "l has been identified in the natural way with In ~l x 0. Thus, / can be aligned provided it has an extension g mapping I" into 7""1 such that g is equivalent to 77. In [2] it was shown that if K is a Cantor set in 72 and/: K-> I is a continuous map such that/-1(0) = F: n (0 x 7) ar\àf-\\) = K n (1 x 7) then/can be aligned. In this paper a corresponding result in higher dimensions is proved. The two main difficulties are that K may hit dln and that, for n ^ 3, K may not be nicely imbedded in In.

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Aligning functions defined on Cantor sets

Ford¹,

Thomas²

1969

Trans. Amer. Math. Soc.

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Extending Real Maps Defined on a Subset of a Disk

Cited by 1 publication

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Aligning functions defined on Cantor sets

Aligning functions defined on Cantor sets

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