On the convergence of the spectrum of Perron-Frobenius operators
For a family \(\{F_t \}_{t= 1,2, \dots, \infty}\) of piecewise \(C^2\) mappings \(F_t: I\to I\) of an interval \(I\) the corresponding Perron-Frobenius operator is defined as follows: \[ \int_I P_t f(x) g(x) dx= \int_I (x) g(F_t (x)) dx. \] Let \(\text{Spec} (F_t)\) denote the spectrum of \(P_t\) restricted to the space BV of functions with a bounded total variation \((V(f)= V(\widetilde {f})< \infty\) for some \(\widetilde {f}- f=0\) in \(L^1\)). If \(F_t\to F_\infty\) in piecewise \(C^1\)-sense, then \(P_t\to P_\infty\) in \(L^1\), but not necessary in BV; thus a general perturbation theory can not be applied. Nevertheless, using the Fredholm matrix, in the main theorem (Theorem A) the author states that \(\text{Spec} (F_\infty)\) can be approximated by \(\text{Spec} (F_t)\).
scientific article; zbMATH DE number 645682