The orbit intersection problem for linear spaces and semiabelian varieties

Let $f_1, f_2 : \mathbb{C}^N \to \mathbb{C}^N$ be affine maps $f_i(x) := A_i x + y_i$ (where each $A_i$ is an $N$-by-$N$ matrix and $y_i \in \mathbb{C}^N$), and let $x_1, x_2 \in \mathbb{A}^N (\mathbb{C})$ 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 $\lbrace (n_1, n_2) \in \mathbb{N}^2_0 : f^{n_1}_1 (x_1) = f^{n_2}_2 (x_2) \rbrace$ is a finite union of sets of the form $\lbrace (m_1 k + \ell_1 , m_2 k + \ell_2) : k \in \mathbb{N}_0 \rbrace$ where $m_1 , m_2, \ell_1 ,\ell_2 \in \mathbb{N}_0$. 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 $\mathbb{C}$ (where $Phi_{i,0} \in \mathrm{End}(X)$ and $y_i \in X(\mathbb{C})$), if none of the eigenvalues of the induced linear action $D \Phi_{i,0}$ on the tangent space at $0 \in X$ is a root of unity (for $i = 1, 2$), then for any two non-preperiodic points $x_1, x_2$, the set $\lbrace (n_1, n_2) \in \mathbb{N}^2_0 : \Phi^{n_1}_1 (x_1) = \Phi^{n_2}_2 (x_2) \rbrace$ is a finite union of sets of the form $\lbrace (m_1 k + \ell_1 , m_2 k +\ell_2) : k \in \mathbb{N}_0 \rbrace$ where $m_1 , m_2, \ell_1, \ell_2 \in \mathbb{N}_0$. 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.

Keywords

dynamical Mordell–Lang problem, intersections of orbits, affine transformations, semiabelian varieties

2010 Mathematics Subject Classification

Primary 11C99, 37P55. Secondary 11B37, 11D61.

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The first author is partially supported by NSERC and the second author is partially supported by a UBC-PIMS postdoctoral fellowship.

Received 14 April 2016

Accepted 3 April 2017

Published 29 December 2017