Abstract and Applied Analysis
Volume 2003 (2003), Issue 2, Pages 75-81
doi:10.1155/S1085337503204012
Local properties of maps of the ball
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
Received 25 January 2002
Copyright © 2003 Yakar Kannai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let f be an essential map of Sn−1 into itself (i.e., f is not homotopic to a constant mapping) admitting an extension
mapping the closed unit ball B¯n into ℝn. Then, for every interior point y of Bn, there exists a point x in f−1(y) such that the image of no neighborhood of x is contained in a coordinate half space with y on its boundary.
Under additional conditions, the image of a neighborhood of x covers a neighborhood of y. Differential versions are valid
for quasianalytic functions. These results are motivated by
game-theoretic considerations.