The Orbit Intersection Problem for Linear Spaces and Semiabelian Varieties

Let f1, f2 : CN −→ CN be affine maps fi(x) := Aix + yi (where each Ai is an N -by-N matrix and yi ∈ CN ), and let x1,x2 ∈ AN (C) such that xi is not preperiodic under the action of fi for i = 1, 2. If none of the eigenvalues of the matrices Ai is a root of unity, then we prove that the set {(n1, n2) ∈ N0 : f n1 1 (x1) = f n2 2 (x2)} is a finite union of sets of the form {(m1k + `1,m2k + `2) : k ∈ N0} where m1,m2, `1, `2 ∈ N0. Using this result, we prove that for any two self-maps Φi(x) := Φi,0(x) + yi on a semiabelian variety X defined over C (where Φi,0 ∈ End(X) and yi ∈ X(C)), if none of the eigenvalues of the induced linear action DΦi,0 on the tangent space at 0 ∈ X is a root of unity (for i = 1, 2), then for any two non-preperiodic points x1, x2, the set {(n1, n2) ∈ N0 : Φ n1 1 (x1) = Φ n2 2 (x2)} is a finite union of sets of the form {(m1k + `1,m2k + `2) : k ∈ N0} where m1,m2, `1, `2 ∈ N0. 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 padic exponential map for semiabelian varieties.