We consider IA-endomorphisms (i.e. Identical in Abelianization) of a free metabelian group of finite rank, and give a matrix characterization of their fixed points which is similar to (yet different from) the well-known characterization of eigenvectors of a linear operator in a vector space. We then use our matrix characterization to elaborate several properties of the fixed point groups of met...

We extend the characterization of virtual permutation endomorphisms in the case where IIi(Ai) = Zn. We show that for endomorphisms of Z[Zn] complexes the appropriate eigenvalue condition is that all eigenvalues be roots of units of the group ring Z[Zn\. Among these endomorphisms the virtual permutations are detected by Kq. The main application is in identifying Morse-Smale isotopy classes on th...

Let S be an abelian group of automorphisms of a probability space (X,A, μ) with a finite system of generators (A1, ..., Ad). Let A l denote A1 1 ...A ld d , for l = (l1, ..., ld). If (Zk) is a random walk on Z , one can study the asymptotic distribution of the sums ∑n−1 k=0 f ◦AZk(ω) and ∑ l∈Z P(Zn = l)A f , for a function f on X . In particular, given a random walk on commuting matrices in SL(...

We consider finite additive cellular automata with fixed and periodic boundary conditions as endomorphisms over pattern spaces. A characterization of the nilpotent and regular parts of these endomorphisms is given in terms of their minimal polynomials. We determine the generalized eigenspace decomposition, and describe relevant cyclic subspaces in terms of symmetries. As an application, we calc...

Suppose Dl and D2 be Riemann surfaces which have bounded nonconstant holomorphic functions. Denote by E ( D i ) , i = 1 , 2 , the semigroups of all holomorphic endomorphisms. If $ : E ( D 1 ) + E(D2) is an isomorphism of semigroups then there exists a conformal or anticonformal isomorphism r/l: Dl 3 D2 such that $ is the conjugation by r/l . Also the semigroup of injective endomorphisms as well...

Let P be a partially ordered set. A function φ : P → P is an endomorphism of P if for every two elements x, y of P , the inequality φ(x) 6 φ(y) holds whenever x 6 y. Obviously, the identity mapping is a (trivial ) endomorphism. Here, however, we will be interested in endomorphisms with an image of size less than |P |, i.e. endomorphisms which are not automorphisms of P . We will refer to them a...