Instability of Isolated Spectrum for W-shaped Maps

TL;DR

This paper demonstrates that the eigenvalue 1 of a specific W-shaped map is unstable under perturbations, leading to metastable behavior with nearly invariant sets and long sojourns in each.

Contribution

It provides a constructive proof of the instability of the eigenvalue 1 for W-shaped maps and shows how perturbations create metastable dynamics.

Findings

Eigenvalue 1 is not stable under perturbations.

Existence of second eigenvalues approaching 1 as perturbation diminishes.

Maps exhibit metastable behavior with long periods in nearly invariant sets.

Abstract

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.