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Instability of the isolated spectrum for W-shaped maps

Published online by Cambridge University Press:  30 May 2012

ZHENYANG LI and
PAWEŁ GÓRA
ZHENYANG LI
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd West, Montreal, Quebec H3G 1M8, Canada (email: zhenyangemail@gmail.com, pgora@mathstat.concordia.ca)
PAWEŁ GÓRA
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd West, Montreal, Quebec H3G 1M8, Canada (email: zhenyangemail@gmail.com, pgora@mathstat.concordia.ca)
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Abstract

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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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013, pp. 1052 - 1059
Copyright
Copyright © 2012 Cambridge University Press 

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