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