On the Convergence of the Spectrum of Perron-Frobenius Operators
Makoto Mori · Tokyo Journal of Mathematics · 1994
Communicated by Y. Ito) 1. Introduction.We will consider $\{F_{t}\}_{t=1,2,\cdots,\infty}$ a family of piecewise $C^{2}$ mappings from an interval $I$ into itself.We denote by $P_{t}$ the Perron-Frobenius operator corresponding to $F_{t}$ :for $f\in L^{1}$ and $g\in L^{\infty}$ , where we denote by $L^{1}$ (resp.$L^{\infty}$ ) the set of integrable functions (resp.the set of bounded measurable functions).We denote by Spec $(F_{t})$ the spectrum of $P_{t}$ restricted to $BV$ , the set of bounded functions.Here, as usual, we consider $BV$ as a subset of $L^{1}$ by taking $L^{1}$ -version and the normWe assume that $F_{t}$ converges to $F_{\infty}$ in piecewise $C^{1}$ (the definition will be stated in \S 2).In this situation, though $P_{t}$ converges to $P_{\infty}$ in $L^{1},$ $P_{t}$ does not necessarily converge to $P_{\infty}$ in $BV$ .This means that general perturbation theories cannot be applied.Nevertheless, using Fredholm matrix which is defined in [10], our main theorem (Theorem A) states that Spec $(F_{\infty})$ can be approximated by Spec $(F_{t})$ .THEOREM A. Assume that (1) each $F_{t}$ is a piecewise $C^{2}$ mapping with positive lower Lyapunov number $\xi_{t}$ $(t=1,2, \cdots, \infty)$ , (2) $F_{t}$ converges to $F_{\infty}$ in piecewise $C^{1}$ .Then for $z_{\infty}$ which $satisfies|z_{\infty}|