Instability of the isolated spectrum for W-shaped maps

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.