From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: What are the Neatest Fixed Point Theorems/Facts to Teach?
Date: 2 Nov 1996 05:18:30 GMT
In article <55djos$n64@oolong.memphis.edu>, David Dwiggins wrote:
>Most FPT's use self-mappings, P:S->S, where P(S) is contained in S.
>But suppose S is contained in P(S); does P have a fixed point in S?
>In the one-dimensional case, the answer is yes (P is of course continuous).
>How about in the plane? Find a continuous P, a compact set S, with S
>contained in P(S), but P having no fixed point in S. I have a solution,
>but it involves P being only continuous on S, not on all of R^2.
P: R^2 -> R^2 a nontrivial rotation, S any annulus around the origin.
Perhaps you want to add the stipulation that S be homeomorphic to the
unit disk?
By the way, if you indeed have a continuous function P : S --> R^2 on a
closed subset S of R^2, the Tietze Extension Theorem may be used to
show that P extends to a continuous map P: R^2 -> R^2.
dave