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Instability of the isolated spectrum for W-shaped maps
Published online by Cambridge University Press: 30 May 2012
Abstract
In this note we consider the W-shaped map $W_0=W_{s_1,s_2}$ with
${1}/{s_1}+{1}/{s_2}=1$ and show that the eigenvalue
$1$ is not stable. We do this in a constructive way. For each perturbing map
$W_a$ we show the existence of a ‘second’ eigenvalue
$\lambda _a$, such that
$\lambda _a\to 1$ as
$a\to 0$, which proves instability of the isolated spectrum of
$W_0$. At the same time, the existence of second eigenvalues close to 1 causes the maps
$W_a$to behave in a metastable way. There are two almost-invariant sets, and the system spends long periods of consecutive iterations in each of them, with infrequent jumps from one to the other.
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- Research Article
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- Copyright
- Copyright © 2012 Cambridge University Press
References
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