Extending real maps defined on a subset of a disk

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

Aligning Functions Defined on Cantor Sets

“…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.…”

Aligning functions defined on Cantor sets

“…The authors thank R. Z. Goldstein and R. P. Osborne for helpful comments during the preparation of this paper. 2. Tameness in I*.…”