Local properties of maps of the ball

Abstract

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 ${ℝ}^n$. Then, for every interior point $y$ of $B^n$, there exists a point $x$ in $f^{-1}\left(y\right)$ 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.