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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On a problem of Nirenberg concerning expanding maps in Hilbert space


Author: Janusz Szczepański
Journal: Proc. Amer. Math. Soc. 116 (1992), 1041-1044
MSC: Primary 47H99; Secondary 47H09
DOI: https://doi.org/10.1090/S0002-9939-1992-1100665-0
MathSciNet review: 1100665
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Abstract: Let $ {\mathbf{H}}$ be a Hilbert space and $ f:{\mathbf{H}} \to {\mathbf{H}}$ a continuous map which is expanding (i.e., $ \vert\vert f({\mathbf{x}}) - f({\mathbf{y}})\vert\vert \geq \vert\vert{\mathbf{x}} - {\mathbf{y}}\vert\vert$ 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}$

$\displaystyle \vert\vert{F^n}({\mathbf{x}}) - {F^n}({\mathbf{y}})\vert\vert \geq {c^{n - m}}\vert\vert{\mathbf{x}} - {\mathbf{y}}\vert\vert$

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 $ \vert\vert F({\mathbf{x}})\vert\vert = c\vert\vert{\mathbf{x}}\vert\vert$ for all $ {\mathbf{x}} \in {\mathbf{H}}$.

We show that no map with the above properties exists in the finite-dimensional case.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1100665-0
Article copyright: © Copyright 1992 American Mathematical Society