On a problem of Nirenberg concerning expanding maps in Hilbert space

Abstract:

Let ${\mathbf {H}}$ be a Hilbert space and $f:{\mathbf {H}} \to {\mathbf {H}}$ a continuous map which is expanding (i.e., $||f({\mathbf {x}}) - f({\mathbf {y}})|| \geq ||{\mathbf {x}} - {\mathbf {y}}||$ for all ${\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}}$) and such that $f({\mathbf {H}})$ has nonempty interior. Are these conditions sufficient to ensure that $f$ is onto? This question was stated by Nirenberg in 1974. In this paper we give a partial negative answer to this problem; namely, we present an example of a map $F:{\mathbf {H}} \to {\mathbf {H}}$ which is not onto, continuous, $F({\mathbf {H}})$ has nonempty interior, and for every ${\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}}$ there is ${n_0} \in \mathbb {N}$ (depending on ${\mathbf {x}}$ and ${\mathbf {y}}$) such that for every $n \geq {n_0}$ \[ ||{F^n}({\mathbf {x}}) - {F^n}({\mathbf {y}})|| \geq {c^{n - m}}||{\mathbf {x}} - {\mathbf {y}}||\] where ${F^n}$ is the $n$th iterate of the map $F,c$ is a constant greater than 2, and $m$ is an integer depending on ${\mathbf {x}}$ and ${\mathbf {y}}$. Our example satisfies $||F({\mathbf {x}})|| = c||{\mathbf {x}}||$ for all ${\mathbf {x}} \in {\mathbf {H}}$. We show that no map with the above properties exists in the finite-dimensional case.