Spectrum Properties of the of the Koopman and Frobenius-Perron operators

Consider the measure space (X,A,m). Let S:X- > X be a non-singular transformation on this space and assume that is the FrobeniusPerron operator associated with s. Also we will assume that Us is the Koopman operator corresponding to s. Properties of this two operators, namely Ps and Us have been considered by many authors. For more details see [5,7,8,9]. In this paper we will present some spacial aspects concerning the spectrum of Ps. Also it is shown that the spectrum of is a cyclic subset of the unit disk D. In connection with the Koopman operator Us it is shown that under certain conditions, if m is a regular measure, then Us from L^& (X) to itself is an isometric.