Variation and Uniqueness of Outchannels
by
John C. Mayer
University of Alabama at Birmingham
Coauthors: Harry Bell (University of Cincinnati), Lex Oversteegen (UAB), Ed Tymchatyn (University of Saskatchewan)

We define Bell's concept of variation of a map of the plane on a subarc of a simple closed curve and relate it to the index of the map on that curve: index = variation + 1. We then define an outchannel for a fixed-point-free map which carries a nonseparating plane continuum into itself, and prove Bell's Lollipop Lemma. We use the latter to show the existence of a unique outchannel, and that the outchannel must have variation = -1. Fixed points theorems for certain special classes of maps follow.

Date received: March 1, 2000