The orbit intersection problem in positive characteristic

In this paper, we study the orbit intersection problem for the linear space and the algebraic group in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic and let $\Phi_1, \Phi_{2}: K^d \longrightarrow K^{d}$ be affine maps, $\Phi_i({\bf x}) = A_i ({\bf x}) + {\bf x_i}$ (where each $A_i$ is a $d\times d$ matrix and ${\bf x}\in K^d$). If none of the eigenvalues of the matrices $A_i$ are roots of unity and each ${\bf a}_i \in K^d$ is not $\Phi_i$-preperiodic, then we prove that the set $\left \{(n_1, n_2) \in \Z^{2} \mid \Phi_1^{n_1}({\bf a}_1) = \Phi_{2}^{n_{2}}({\bf a}_{2})\right\}$ is $p$-normal in $\mathbb{Z}^{2}$ of order at most $d$. Further, let $\Phi_1, \Phi_{2}: \mathbb{G}_m^d \longrightarrow \mathbb{G}_m^d$ be regular self-maps and ${\bf a}_1, {\bf a}_2\in \mathbb{G}_m^d(K)$. Let $\Phi_1^0$ and $\Phi_2^0$ be group endomorphisms of $\mathbb{G}_m^d$ and ${\bf y}, {\bf z} \in \mathbb{G}_m^d(K)$ such that $\Phi_1({\bf x}) = \Phi_1^{0}({\bf x}) + {\bf y}$ and $\Phi_2({\bf x}) = \Phi_2^{0}({\bf x}) + {\bf z}$. We show, under some conditions on the roots of the minimal polynomial of $\Phi_1^{0}$ and $\Phi_2^{0}$, that the set $\{(n_1, n_{2}) \in \N_0^{2} \mid \Phi_1^{n_1}({\bf a}_1) = \Phi_{2}^{n_{2}}({\bf a}_{2})\}$ (where ${\bf a}_1, {\bf a}_2\in \mathbb{G}_m^d(K)$) is a finite union of singletons and one-parameter linear families. To do so, we use results on linear equations over multiplicative groups in positive characteristic and some results on systems of polynomial-exponential equations.