Local properties of maps of the ball (Q1864357)

Summary: Let \(f\) be an essential map of \(S^{n-1}\) into itself (i.e., \(f\) is not homotopic to a constant mapping) admitting an extension mapping the closed unit ball \(\overline B^n\) into \(\mathbb{R}^n\). Then, for every interior point \(y\) of \(B^n\), 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.