Instability of Isolated Spectrum for W-shaped Maps

Details Instability of Isolated Spectrum for W-shaped Maps Instability of Isolated Spectrum for W-shaped Maps

2011-10-16

arxiv.org/abs/1110.3528v1

Mathematics

Dynamical Systems

12 pages, 2 figures

Scientific paper

In this note we consider $W$-shaped map $W_0=W_{s_1,s_2}$ with $\frac {1}{s_1}+\frac {1}{s_2}=1$ and show that eigenvalue 1 is not stable. We do this in a constructive way. For each perturbing map $W_a$ we show the existence of the "second" eigenvalue $\lambda_a$, such that $\lambda_a\to 1$, as $a\to 0$, which proves instability of isolated spectrum of $W_0$. At the same time, the existence of second eigenvalues close to 1 causes the maps $W_a$ behave in a metastable way. They have 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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