The orbit intersection problem for linear spaces and semiabelian varieties

Let f(1), f (2) : C-N -> C-N be affine maps f(i) (x) := A(i)x + y(i) (where each A(i) is an N-by-N matrix and y(i) is an element of C-N), and let x(1), x(2) is an element of A(N) (C) such that x(i) is not preperiodic under the action of f(i) for i = 1, 2. If none of the eigenvalues of the matrices A(i) is a root of unity, then we prove that the set {(n(1), n(2)) is an element of N-0(2) :f(1)(n1) (x(1)) = f (n2)(2) (x(2))} is a finite union of sets of the form {(m(1)k + l(1), m(2)k + l(2)) : k is an element of N-0} where m(1), m(2), l(1), l(2) is an element of N-0. Using this result, we prove that for any two self-maps Phi(i) (x) := Phi(i),(0) (x) + y(i) on a semiabelian variety X defined over C (where Phi,(0) is an element of End(X) and y(i) is an element of X (C)), if none of the eigenvalues of the induced linear action D Phi(i,0) on the tangent space at 0 is an element of X is a root of unity (for i = 1, 2), then for any two non-preperiodic points x(1), x(2), the set {(n(1), n(2)) is an element of N-0(2) : Phi(n1)(1) (x(1)) = Phi(n2)(2) (x(2))} is a finite union of sets of the form {(m(1)k + l(1), m(2)k + l(2)) : k is an element of N-0} where m(1), m(2), l(1), l(2) is an element of N-0. We give examples to show that the above condition on eigenvalues is necessary and introduce certain geometric properties that imply such a condition. Our method involves an analysis of certain systems of polynomial-exponential equations and the p-adic exponential map for semiabelian varieties.