## Abstract and Applied Analysis

### Local properties of maps of the ball

Yakar Kannai

#### 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 $\mathbb{R}^n$. 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2003, Number 2 (2003), 75-81.

Dates
First available in Project Euclid: 15 April 2003

https://projecteuclid.org/euclid.aaa/1050426052

Digital Object Identifier
doi:10.1155/S1085337503204012

Mathematical Reviews number (MathSciNet)
MR1960138

Zentralblatt MATH identifier
1017.26020

Subjects
Primary: 26E10: $C^\infty$-functions, quasi-analytic functions [See also 58C25] 58K05
Secondary: 55M25 47H10 47H11 57N75 57Q65

#### Citation

Kannai, Yakar. Local properties of maps of the ball. Abstr. Appl. Anal. 2003 (2003), no. 2, 75--81. doi:10.1155/S1085337503204012. https://projecteuclid.org/euclid.aaa/1050426052