Instability of the isolated spectrum for W-shaped maps (Q2842226)

Let \(s_1,s_2>1\) with \(1/s_1 + 1/s_2 = 1\). The W-shaped map \(W_{s_1,s_2}\) is the piecewise linear self-map of the interval \([0,1]\) whose graph looks like a W, where the first and last branches are onto \([0,1]\), the second branch has slope \(s_1\), the third branch has slope \(-s_2\), and the second and third branch meet at \(W_{s_1,s_2}(1/2) = 1/2\). The absolutely continuous \(W_{s_1,s_2}\)-invariant measures on \([0,1]\) correspond to the eigenfunctions with eigenvalue \(1\) of the Perron-Frobenius operator (transfer operator) associated to \(W_{s_1,s_2}\).NEWLINENEWLINEThe authors show that the eigenvalue \(1\) is not stable under small perturbations of \(W_{s_1,s_2}\). The proof is constructive: They provide a family \((W_a)_{a>0}\) of W-shaped maps with \(W_0 = W_{s_1,s_2}\) and \(W_a(1/2) = 1/2 + a\) such that, for \(a>0\), the Perron-Frobenius operator has a second eigenvalue \(\lambda_a<1\). This causes metastability for \(a>0\) and instability in the limit \(a\to 0\).