The orbit intersection problem for linear spaces and semiabelian varieties

TL;DR

This paper characterizes the intersection patterns of orbits under affine maps and self-maps on semiabelian varieties, showing they form structured sets under certain eigenvalue conditions, with implications for polynomial-exponential equations.

Contribution

It extends orbit intersection results to affine maps and semiabelian varieties, establishing finiteness and structure of intersection sets based on eigenvalue conditions.

Findings

The intersection set is a finite union of arithmetic progressions under eigenvalue conditions.

Eigenvalues not being roots of unity are necessary for the structured intersection property.

The method involves polynomial-exponential equations and p-adic exponential maps.

Abstract

Let f_1 and f_2 be affine maps of the N-th dimensional affine space over the complex numbers, i.e., f_i(x):=A_i x + y_i (where each A_i is an N-by-N matrix and y_i is a given vector), and let x_1 and x_2 be vectors 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 of pairs (n_1,n_2) of non-negative integers such that f_1^{n_1}(x_1)=f_2^{n_2}(x_2) is a finite union of sets of the form (m_1k + \ell_1, m_2k + \ell_2) where m_1, m_2, \ell_1, \ell_2 are given non-negative integers, and k is varying among all non-negative integers. 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 the complex numbers (where \Phi_{i,0} is an endomorphism of X and y_i is a given point of X), if none of the eigenvalues of the induced linear action D\Phi_{i,0} on the tangent space at the identity 0 of X is a root of unity (for i=1,2), then for any two non-preperiodic points x_1,x_2, the set of pairs (n_1,n_2) of non-negative integers such that \Phi_1^{n_1}(x_1) = \Phi_2^{n_2}(x_2) is a finite union of sets of the form (m_1k + \ell_1, m_2k + \ell_2) where m_1,m_2,\ell_1,\ell_2 are given non-negative integers, and k is varying among all non-negative integers. 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.