Positive linear maps are very interesting objects in operator algebra theory, and have been studied by many authors (cf. [ 1,3, 151). In recent years, the theory of dynamical semi-groups, which are defined as certain semi-groups of completely positive maps, has made remarkable progress (cf. [6,8,11]). The study of such semi-groups was initiated by physicists as one of the natural mathematical frameworks for quantum theory of open systems, and so the investigations have been mainly made under rather strong restrictions which are physically justified. On the other hand, we wish to study such semi-groups from the mathematical point of view, thus we shall consider more genera1 semigroups; that is, we define a dynamical semi-group a = {a,)r>o as a u-weakly continuous one-parameter semi-group of normal positive maps on a von Neumann algebra M. In [ 171, we have shown various ergodic theorems for such a semi-group under the assumption that it possesses an invariant faithful normal state p. The triplet (M, a, p) is called a dynamical system. The purpose of this paper is to investigate the relationship between the asymptotic behavior (such as ergodicity or mixing properties) of (M, a,p) and eigenvalues of a. In Section 2, we present basic notations and terminologies, and consider dual dynamical systems. In Section 3, we define eigenvalues of a and examine their basic properties. In particular, we show that for any real eigenvalues 1 of a there exists a a-weakly continuous norm one projection onto the eigenspace MA under the additional assumption that every a, is 2-positive. In Section 4, eigenspaces of products of dynamical semi-groups are computed. In Section 5, we show that ergodicity of (M, a,p) is equivalent to extremality of p in the set of all a-invariant states of M, and is also equivalent to that p is only a a-invariant normal state. Finally, in Section 6, mixing properties are characterized by using various conditions. Especially, we show that (M, a,p) is weakly mixing if and only if it is 411 0022.247X/82!;0404li-14$02.00.0