Mathematical Research Letters

Volume 24 (2017)

Number 5

The orbit intersection problem for linear spaces and semiabelian varieties

Pages: 1263 – 1283

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n5.a2

Authors

Dragos Ghioca (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)

Khoa Dang Nguyen (Department of Mathematics and Statistics, University of Calgary, Alberta, Canada)

Abstract

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.

Full Text (PDF format)

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

Published 29 December 2017