The orbit intersection problem for linear spaces and semiabelian varieties

Abstract

Abstract. 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 ∈ C N ), and let x 1 , x 2 ∈ 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 ) ∈ N 2 0 : f2 (x 2 )} is a finite union of sets of the formUsing this result, we prove that for any two self-maps Φ i (x) := Φ i,0 (x) + y i on a semiabelian variety X defined over C (where Φ i,0 ∈ End(X) and y i ∈ X(C)), if none of the eigenvalues of the induced linear action DΦ i,0 on the tangent space at 0 ∈ 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 ) ∈ N 2 0 : Φ2 (x 2 )} is a finite union of sets of the formWe 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.

Unit equations on quaternions

“…Analogous results have been proven in various cases in characteristic zero, in case E is replaced by A 1 [8], a linear space [6], or a semiabelian variety [6,7]. Furthermore, the analogue of Corollary 1.5 to arbitrary simple abelian varieties A would follow from the proof of Corollary 1.5, if one could prove the relevant case of Conjecture 1.4 for the division algebras End(A).…”

“…The characteristic zero case of Corollary 1.5 is an instance of the higherrank generalization posed in [8, Question 1.6] of the dynamical Mordell-Lang conjecture [2, Chapter 3]; see also [6]. For positive characteristic, see [2,Chapter 13].…”

A new mean ergodic theorem for tori and recurrences

“…Let N denote the set of positive integers and let N 0 = N ∪ {0}. The proof of Theorem 1.2 is motivated by the theorem of Ghioca (below) and the author [GN17], in arithmetic dynamics.…”

The orbit intersection problem in positive characteristic

“…In this paper, we study Question 1.1 in positive characteristic. From the following example (see [11]), one can see that Question 1.1 fails for affine maps defined over F p (t). Let Φ i : A 1 −→ A 1 be affine maps defined by Φ 1 (x) = t(x − 1) + 1 and Φ 2 (x) = (t + 1)x.…”

“…Question 1.1 is known for the case when X = P 1 K and each Φ i is a polynomial of degree larger than 1 (see [13,15]). Later, Question 1.1 is answered when X is a semiabelian variety and when X = A n K and the self maps are affine transformations [11]. Further in [20], various upper bounds are derived for the orbit intersection problem when X is an affine n-space and self maps are polynomial morphisms of special types.…”