Abstract and Applied Analysis
Volume 2003 (2003), Issue 2, Pages 75-81

Local properties of maps of the ball

Yakar Kannai

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.


Let f be an essential map of Sn1 into itself (i.e., f is not homotopic to a constant mapping) admitting an extension mapping the closed unit ball B¯n into n. Then, for every interior point y of Bn, there exists a point x in f1(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.