The Spectral Analysis of Frobenius-Perron Operators

It is known that the Frobenius-Perron operator Ps:L1(0,1)→L1(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 L1(0,1), and thus Ps: BV[0,1]→BV[0,1] possesses only the finite peripheral spectrum and in particular 1 is an isolated eigenvalue of Ps. In this paper, we show that under mild conditions on S, the spectrum of Ps:L1(X)→L1(X) is either the closed unit disk {λϵC:|λ|≤1} or a cyclic subset of {λϵC:|λ|=1}.