The spectral analysis of Frobenius-Perron operators
It is known that the Frobenius-Perron operator \(P_S: L^1(0, 1)\to L^1(0, 1)\) associated with a transformation \(S\) from \([0, 1]\) to itself with \(\inf|S'|> 1\) is quasi-compact as an operator on the Banach space \(BV[0, 1]\) of functions of bounded variation in \(L^1(0, 1)\), and thus \(P_S: BV[0, 1]\to BV[0, 1]\) possesses only the finite peripheral spectrum and in particular \(1\) is an isolated eigenvalue of \(P_S\). In this paper, the authors show that under mild conditions on \(S\), the spectrum of \(P_S: L^1(X)\to L^1(X)\) is either the closed unit disk \(\{\lambda\in \mathbb{C}: |\lambda|\leq 1\}\) or a cyclic subset of \(\{\lambda\in \mathbb{C}: |\lambda|= 1\}\).
scientific article; zbMATH DE number 635924