  Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in
GL_n(Z). Assume that the characteristic polynomial of M is prime to all
polynomials X^m-1. We show that any finite equivariant map from another
algebraic dynamics onto (T,M) arises from a finite isogeny T \to T. A similar
and more general statement is shown for Abelian and semi-abelian varieties.
  In model-theoretic terms, our result says: Working in an existentially closed
difference field, we consider a definable subgroup B of a semi-abelian variety
A; assume B does not have a subgroup isogenous to A'(F) for some twisted fixed
field F, and some semi-Abelian variety A'. Then B with the induced structure is
stable and stably embedded. This implies in particular that for any n>0, any
definable subset of B^n is a Boolean combination of cosets of definable
subgroups of B^n.
  This result was already known in characteristic 0 where indeed it holds for
all commutative algebraic groups ([CH]). In positive characteristic, the
restriction to semi-abelian varieties is necessary.
