2·1. Matrices over skew fields

Fact 2·1. Let K be a skew field with centre k and $A \in M_n(K)$ be a matrix with a minimal polynomial equal to $(x - \alpha)^r$ for some $\alpha \in k$ and $r \in {\mathbb N}$ . Then, there exist an invertible matrix $P \in M_n(K)$ such that

\begin{equation*}P^{-1}AP = J_{\alpha, r_1} \bigoplus \cdots \bigoplus J_{\alpha, r_m},\end{equation*}

where $J_{\alpha,s}$ is the s-by-s Jordan canonical matrix having unique eigenvalue $\alpha$ and its only nonzero entries away from the diagonal being the entries in positions $(i,i+1)$ (for $i=1,\dots, s-1$ ), which are all equal to 1.

2·2. Semiabelian varieties

We recall that a semiabeian variety G defined over an algebraically closed field L is an algebraic group variety, which is an extension of an abelian variety A by a torus ${\mathbb G}_m^N$ :

(2·1) \begin{equation}1\longrightarrow {\mathbb G}_m^N\longrightarrow G\longrightarrow A\longrightarrow 1.\end{equation}

We say that G is split if the short exact sequence of algebraic groups from (2·1) splits. In this case, G is isogenous to a product of simple semiabelian varieties (i.e., semiabelian varieties which contain no proper semiabelian varieties).

As previously noted (see [ Reference Iitaka16 ]), any regular self-map on a semiabelian variety G is a composition of a group endomorphism of G with a translation map $\tau_{\beta}$ (where $\tau_\beta(x)=x+\beta$ for each $x\in G$ ).

Definition 2·3. We define a split semiabelian variety G to be reduced if G is isomorphic to

(2·2) \begin{equation} \prod_{i = 1}^rC_i^{k_i},\end{equation}

where $k_1, \dots, k_r \in {\mathbb N}$ and $C_1, \dots,C_r$ are simple semiabelian varieties that are pairwise non-isogenous.

2·3. Correspondences on semiabelian varieties

For a semiabelian variety G, a correspondence or finite-to-finite map is a dominant map $\varphi\in {1}/{m}\cdot {\rm End}(G)$ for some $m\in{\mathbb N}$ . In other words, there exists a positive integer m such that composing the multiplication-by-m map $[m]_G$ on G with $\varphi$ yields a well-defined, dominant endomorphism of G. Clearly, for each point $\alpha\in G$ , we have that $\varphi(\alpha)$ consists of at most $m^{2\dim(G)}$ points, which all differ by a torsion point of G of order dividing m.

2·4. Almost commutative diagrams

We call an almost commutative diagram, a diagram of the following form:

(2·3)

where G and G ^′ are semiabelian varieties, $g\;:\;G'\longrightarrow G$ is an isogeny, $\Psi\;:\;G'\longrightarrow G'$ is a group endomorphism, while $\Phi\;:\;G\longrightarrow G$ is a correspondence such that there exists a positive integer $m_0$ for which

(2·4) \begin{equation}[m_0]_G\circ g\circ \Psi = [m_0]_G\circ \Phi\circ g.\end{equation}

In particular, we have that $[m_0]_G\circ \Phi$ is a well-defined, regular endomorphism of G. Furthermore, letting $\hat{g}\;:\;G\longrightarrow G'$ be an isogeny such that

\begin{equation*}\hat{g}\circ g=[m]_{G'}\text{ and }g\circ \hat{g}=[m]_G,\end{equation*}

for some positive integer m divisible by $m_0$ , we obtain:

(2·5) \begin{equation}[m]_G\circ\Phi = g\circ \Psi\circ \hat{g}.\end{equation}

Furthermore, since $m_0$ divides m, we have that $[m]_G\circ\Phi$ from (2·5) is a well-defined endomorphism. In particular, we see that we can take $m_0\;:\!=\;m$ in equation (2·4) and thus, $\Phi\in {1}/{m}\cdot {\rm End}(G)$ is defined through equation (2·5).

2·5. Some technical definitions and notation relevant to our proofs

We will also employ the following notation regarding the endomorphisms of a (simple) semiabelian variety.

Next definition is motivated by condition (c) in Theorem 1·3.

2·6. The intersection of a subvariety of a semiabelian variety defined over a finite field with a finitely generated subgroup

We conclude this technical background section with stating the F-structure theorem of Moosa–Scanlon for the intersection of a subvariety of an $\mathbb F_q$ -semiabelian variety G with a finitely generated subgroup of G(K) (where K is some arbitrary algebraically closed field containing $\mathbb F_q$ ). In order to state Theorem 2·8 (which is an essential ingredient for our proofs), we need first to introduce the notion of F-sets defined by Moosa–Scanlon [ Reference Moosa and Scanlon21 ]. The Frobenius F acting on G is the endomorphism induced by the usual field homomorphism given by $x\mapsto x^q$ for each $x\in K$ .

We note that since we allow the base points $a_i$ be outside $\Gamma$ , we can use the slightly simpler definition of groupless F-sets involving sums of F-orbits rather than using the F-cycles (see [ Reference Moosa and Scanlon21 , remark 2·6], and also the extension proven in [ Reference Ghioca7 ]). We also refer to [ Reference Corvaja, Ghioca, Scanlon and Zannier5 , section 2·2] for a more in-depth discussion of the structure of F-sets.

Remark 2·9. Furthermore, according to [ Reference Moosa and Scanlon21 , remark 2·6], if $\Gamma$ is a finitely generated ${\mathbb Z}[F]$ -submodule of G(K), then the F-sets appearing in the intersection $V\cap\Gamma$ from Theorem 2·8 are of the form $C(\gamma,\alpha_1,\dots,\alpha_m;\; k_1,\dots,k_m)+\Gamma '$ (see Definition 2·7) where for some positive integer $\ell$ , we have that

\begin{equation*}\ell\cdot \gamma,\ell\cdot\alpha_1,\dots, \ell\cdot \alpha_m\in \Gamma.\end{equation*}

Finally, in our proof, we prefer to use the notation

\begin{equation*}\Sigma(\alpha_1,\dots,\alpha_m;\;k_1,\dots,k_m)\;:\!=\;\left\{\sum_{j=1}^m F^{k_jn_j}(\alpha_j) \colon n_j\in\mathbb{N}_0\right\}\end{equation*}

for a sum of F-orbits (for given points $\alpha_j\in G(K)$ and positive integers $k_j$ ).

3·1. Reduction to the case the roots of $g_\psi$ are not roots of unity of order greater than 1

We first note the following reduction in Theorem 1·3.

After replacing $\Psi$ by a suitable iterate (see Proposition 3·1 and also formula (3·1)) we may assume without loss of generality that the roots of the minimal polynomial of $\psi$ (over ${\mathbb Z}$ ) that are roots of unity are actually all equal to one.

3·2. Writing the minimal polynomial of $\psi$ as a product of two coprime polynomials with special properties

Let $g\;:\!=\;g_\psi\in {\mathbb Z}[x]$ be the minimal polynomial for the endomorphism $\psi$ . As explained in the previous section, we may assume that each root of g is either equal to 1 or not a root of unity.

We let $s\in{\mathbb N}_0$ be the order of 1 as a root of g(x). We write $h_1(x)\;:\!=\;(x-1)^s$ ; then we can write $g(x) \;:\!=\; h_1(x) \cdot h_2(x)$ for some polynomial $h_2(x)$ with integer coefficients whose roots are not roots of unity. Furthermore, $h_1(x)$ and $h_2(x)$ are coprime polynomials.

3·3. Splitting the action of $\Psi$ to an action on a product of two special semiabelian varieties

We continue with the notation for $h_1(x)$ and $h_2(x)$ from Section 3·2 and we let $G_1 \;:\!=\; h_2(\psi)\big(G\big)$ and $G_2 \;:\!=\; h_1(\psi)\big(G\big)$ . Then $G_1$ and $G_2$ are both connected algebraic subgroups of G (note that either $G_1$ or $G_2$ may be the trivial group). Since $h_1$ and $h_2$ are coprime, then there exist polynomials with integer coefficients $Q_1(x)$ and $Q_2(x)$ along with some positive integer $\ell_0$ such that

\begin{equation*}Q_1(x)\cdot h_1(x) + Q_2(x)\cdot h_2(x) = \ell_0,\end{equation*}

which means that $G_1$ and $G_2$ are complementary subgroups of G, in the sense that $ G = G_1 + G_2$ , while $G_1\cap G_2$ is finite (consisting only of points of order dividing $\ell_0$ ). Thus, for each $x\in G$ one can find $x_1 \in G_1$ and $x_2\in G_2$ such that $x=x_1+x_2$ ; even though $x_1$ and $x_2$ are not uniquely defined by x, since $G_1\cap G_2$ consists only of points of order dividing $\ell_0$ , we conclude that the isogeny $\iota\;:\;G\longrightarrow G_1\times G_2$ given by

(3·2) \begin{equation}x\longmapsto (\ell_0x_1,\ell_0x_2)\text{ is well-defined.}\end{equation}

Furthermore, $\psi$ induces endomorphisms of both $G_1$ and $G_2$ ; call them $\psi_1$ , respectively $\psi_2$ . In addition,

(3·3) \begin{equation}\text{the minimal polynomial of $\psi_1$ is $h_1(x)=(x-1)^s$,}\end{equation}

(3·4) \begin{equation}\text{while the minimal polynomial of $\psi_2$ is $h_2(x)$.}\end{equation}

Since $G_1+G_2=G$ , then there exist $\beta_1\in G_1(K)$ and $\beta_2\in G_2(K)$ such that $\beta_1+\beta_2=\beta$ . Furthermore, according to (3·2), regardless of our choice of $(\beta_1,\beta_2)\in G_1\times G_2$ for which $\beta_1+\beta_2=\beta$ , we have that the pair $(\ell_0\beta_1,\ell_0\beta_2)$ is unchanged.

Now, we define $\Psi_1\;:\;G_1\longrightarrow G_1$ and $\Psi_2\;:\;G_2\longrightarrow G_2$ given by

(3·5) \begin{equation}\Psi_i(x)=\psi_i(x)+\ell_0\beta_i\text{ for }i=1,2.\end{equation}

Then, using the isogeny $\iota$ (see (3·2)) along with the definition of $\Psi_1$ and $\Psi_2$ (see (3·5)), we have that the following diagram commutes

(3·6)

3·4. Reduction of the action of $\psi_1$ to an endomorphism of a reduced split semiabelian variety

Since $G_1$ is a semiabelian subvariety of G, then also $G_1$ is a split semiabelian variety and must be isogenous to a semiabelian variety

(3·7) \begin{equation}G'_{\!\!1} \;:\!=\; \prod_{i=1}^r C_i^{k_i},\end{equation}

where the $C_i$ ’s are non-isogenous simple semiabelian varieties. More precisely, we have an isogeny

(3·8) \begin{equation}\pi \;:\; G_1 \longrightarrow G'_{\!\!1}\end{equation}

and another isogeny

(3·9) \begin{equation}\hat{\pi} \;:\; G'_{\!\!1}\longrightarrow G_1\end{equation}

along with some positive integer $m_1$ such that

(3·10) \begin{equation}\pi\circ \hat{\pi} = [m_1]_{G'_{\!\!1}}, \quad \hat{\pi} \circ \pi = [m_1]_{G_1}.\end{equation}

Consider $\varphi'_{\!\!1} \in {\rm End}(G'_{\!\!1}) \otimes {\mathbb Q}$ given by

(3·11) \begin{equation}\varphi'_{\!\!1} \;:\!=\; \frac{1}{m_1}\pi\circ \psi_1\circ \hat{\pi}.\end{equation}

Then $\varphi'_{\!\!1}$ corresponds to a direct sum $A_1 \oplus \cdots \oplus A_r$ of matrices in

\begin{equation*}\prod_{i=1}^r M_{k_i}\left({\rm End}(C_i)^0\right).\end{equation*}

Furthermore, since the minimal polynomial of each $A_i$ over ${\mathbb Z}$ is of the form $(x-1)^{s_i}$ for some integer $s_i\le s$ (see (3·3)), then each $A_i$ is a unipotent matrix. So, using Fact 2·1, there exist matrices $P_i \in M_{k_i}({\rm End}(C_i)^0)$ such that $P_iA_i P_i^{-1}$ is of the form

\begin{equation*}B_i = \bigoplus_{j=1}^{\ell_i} J_{1, m_j^{(i)}}\end{equation*}

for some $\ell_i\in{\mathbb N}$ and some positive integers $m_j^{(i)}$ such that

\begin{equation*}\sum_{j=1}^{\ell_i}m_j^{(i)}=k_i.\end{equation*}

This means (see Section 2·4) that there must exist $\sigma, \hat{\sigma}\;:\; G'_{\!\!1} \longrightarrow G'_{\!\!1}$ such that

(3·12) \begin{equation}\sigma \circ \hat{\sigma} = \hat{\sigma} \circ \sigma = [m'_{\!\!1}]_{G'_{\!\!1}}\end{equation}

for some $m'_{\!\!1} \in {\mathbb N}$ and

(3·13) \begin{equation}\varphi_1 = \frac{1}{m'_{\!\!1}}\sigma \circ \varphi'_{\!\!1} \circ \hat{\sigma},\end{equation}

where $\varphi_1$ is the endomorphism corresponding to $B_1\oplus \cdots \oplus B_r$ .

3·5. Reducing to the case $\Psi_2$ is a group endomorphism

Now, since the minimal polynomial of $\psi_2$ (which is $h_2(x)$ , according to (3·4)) does not have any roots that are equal to one, then we have that $\psi_2-{\rm id}_{G_2}$ is an dominant group endomorphism of $G_2$ and therefore, we can find $z\in G_2(K)$ such that

(3·14) \begin{equation}\left(\psi_2-{\rm id}_{G_2}\right)(z)=\ell_0\beta_2.\end{equation}

So, letting $\tau\;:\;G_2\longrightarrow G_2$ be the translation-by-z map, then (3·14) yields that

(3·15) \begin{equation}\psi_2 \;:\!=\;\tau\circ \Psi_2\circ \tau^{-1}\text{ is an endomorphism of }G_2.\end{equation}

3·6. Reducing the dynamical system on G to a simpler dynamical system on $G'_{\!\!1}\times G_2$

We let $\nu\;:\!=\;\sigma\circ \pi$ and using (3·10), (3·11), (3·12) and (3·13), we get that

(3·16) \begin{equation}\nu\circ \psi_1=\varphi_1\circ \nu.\end{equation}

We let $\Phi_1\;:\;G'_{\!\!1}\longrightarrow G'_{\!\!1}$ given by $\Phi_1(x)=\varphi_1(x)+\nu(\ell_0\beta_1)$ . Since $\Psi_1\;:\;G_1\longrightarrow G_1$ is given by $\Psi_1(x)=\psi_1(x)+\ell_0\beta_1$ (see (3·5)), then we conclude that

(3·17) \begin{equation}\nu\circ \Psi_1=\Phi_1\circ \nu.\end{equation}

So, letting $g_1 \;:\!=\; (\nu, \tau)$ and also using (3·15) and (3·17), then we get the next commutative diagram

(3·18)

3·7. Deconstructing the action of $\psi_2$ on $G_2$ using correspondences

Since $G_2$ is a semiabelian subvariety of a split semiabelian variety, then also $G_2$ is isogenous to a reduced split semiabelian variety

(3·19) \begin{equation}G'_{\!\!2} \;:\!=\; \prod_{i=1}^{r^{\prime}} \left(C'_{\!\!i}\right)^{k'_{\!\!i}}.\end{equation}

Moreover, one can choose the components $C'_{\!\!i}$ so that for any $1 \le i \le r^{\prime}$ and $1 \le j \le r$ , $C'_{\!\!i}$ is isogenous to $C_j$ if and only if $C'_{\!\!i} = C_j$ . In other words, the simple semiabelian components of $G'_{\!\!1}$ and $G'_{\!\!2}$ are either equal or they are non-isogenous. So, we have isogenies

(3·20) \begin{equation}\pi' \;:\; G_2 \longrightarrow G'_{\!\!2}, \quad \hat{\pi}' \;:\; G'_{\!\!2}\longrightarrow G_2\end{equation}

along with some positive integer $n_1$ such that

(3·21) \begin{equation}\pi'\circ \hat{\pi}' = [n_1]_{G'_{\!\!2}}, \quad \hat{\pi}' \circ \pi' = [n_1]_{G_2}.\end{equation}

Now, consider $\varphi'_{\!\!2}\in {\rm End}(G'_{\!\!2})\otimes {\mathbb Q}$ given by

(3·22) \begin{equation}\varphi'_{\!\!2} \;:\!=\; \frac{1}{n_1}\pi'\circ (\psi_2)\circ \hat{\pi}'.\end{equation}

Then $\varphi'_{\!\!2}$ is a finite-to-finite map (or correspondence), i.e., it sends any finite subset of $G'_{\!\!2}$ into another finite subset of $G'_{\!\!2}$ . Also, we see that $\varphi'_{\!\!2}$ can be represented naturally in

(3·23) \begin{equation}\prod_{i=1}^{r^{\prime}} M_{k'_{\!\!i}, k'_{\!\!i}}\left(\frac{1}{n_1} D_i\right),\end{equation}

where $D_i \;:\!=\; {\rm End}(C'_{\!\!i})$ , while $D_{i}/{n_1}$ means that we allow denominator $n_1$ for each entry in the corresponding matrices. We also fix an embedding of each ${\mathbb Q}(F_{C'_{\!\!i}})$ (for $i=1,\dots, r$ ) into $\overline{{\mathbb Q}}$ .

3·8. Linearising the action of $\varphi'_{\!\!2}$ on $G'_{\!\!2}$

The action of $\varphi'_{\!\!2}\in{\rm End}(G'_{\!\!2})\otimes {\mathbb Q}$ corresponds to a direct sum of matrices

\begin{equation*}\tilde{A}_{\varphi^{\prime}_{2}}\;:\!=\;A'_{\!\!1} \oplus \cdots \oplus A'_{\!\!r^{\prime}} \in \prod_{i=1}^{r^{\prime}} M_{k'_{\!\!i}, k'_{\!\!i}}\left(\frac{1}{n_1} D_i\right).\end{equation*}

3·9. The minimal polynomial of $\tilde{A}_{\varphi^{\prime}_{2}}$

Using equations (3·21) and (3·22), we see that for each $n\in{\mathbb N}$ , we have that

(3·24) \begin{equation}(\varphi'_{\!\!2})^n \;:\!=\; \frac{1}{n_1}\pi'\circ (\psi_2)^n\circ \hat{\pi}'\end{equation}

and therefore, the minimal polynomial for $\tilde{A}_{\varphi^{\prime}_{2}}$ (which is the matrix in ${\rm End}(G'_{\!\!2})\otimes{\mathbb Q}$ corresponding to $\varphi'_{\!\!2}$ ) is the same as the minimal polynomial of $\psi_2$ as an endomorphism of $G'_{\!\!2}$ . Furthermore, using (3·15) along with (3·4), we conclude that the minimal polynomial for $\tilde{A}_{\varphi^{\prime}_{2}}$ is $h_2(x)$ .

Remark 3·2. Equation (3·24) also yields that for each $x\in G'_{\!\!2}$ , we have that for any $n\in{\mathbb N}$ and for any two points $y,z\in (\varphi'_{\!\!2})^n(x)$ (i.e., for any two points y and z associated to x by the correspondence $(\varphi'_{\!\!2})^n$ ), we have that

(3·25) \begin{equation}y-z\in G'_{\!\!2}[n_1]\text{ (i.e., it is a torsion point of order dividing $n_1$).}\end{equation}

3·10. Separating the roots of $h_2(x)$

As shown in the previous Section, we know that the minimal polynomial of the matrix $\tilde{A}_{\varphi^{\prime}_{2}}$ is $h_2(x)$ and since $h_2(x)\in{\mathbb Z}[x]$ is a monic polynomial, we conclude that

(3·26) \begin{equation}\text{each root of $h_2(x)$ is integral over ${\mathbb Z}$.}\end{equation}

Using Proposition 3·1, we can replace $\Psi$ by a suitable iterate (which leads to replacing $\varphi'_{\!\!2}$ by a corresponding iterate and therefore, replacing each matrix $A'_{\!\!i}$ by its suitable power), so that we may assume that the roots of the minimal polynomial of each $A'_{\!\!i}$ over ${\mathbb Q}\left(F_{C'_{\!\!i}}\right)$ are either a power of $F_{C'_{\!\!i}}$ or multiplicatively independent with respect to $F_{C'_{\!\!i}}$ . Furthermore, writing each such multiplicatively dependent root of the minimal polynomial of $A'_{\!\!i}$ as $F_{C'_{\!\!i}}^{n_j^{(i)}}$ for some integer $n_j^{(i)}$ (where $1\le j\le s_i$ for some $s_i\in{\mathbb N}_0$ ), we note that the exponents $n_j^{(i)}$ must be positive integers because we know the roots of $h_2(x)$ are not roots of unity and also, we know that these roots must be integral over ${\mathbb Z}$ , according to (3·26).

3·11. Splitting the action of $\tilde{A}_{\varphi^{\prime}_{2}}$ into a suitable direct product

Using Facts 2·1 and 2·2 along with the notation from the previous section regarding the roots of each minimal polynomial of $A'_{\!\!i}$ as being either of the form $F_{C'_{\!\!i}}^{n_j^{(i)}}$ for some $n_j^{(i)}\in{\mathbb N}$ (where $1\le j\le s_i$ ) or being multiplicatively independent with respect to $F_{C'_{\!\!i}}$ , there must exist matrices $P'_{\!\!i} \in M_{k'_{\!\!i}}\left({\rm End}(C'_{\!\!i})^0\right)$ such that

(3·27) \begin{equation}P'_{\!\!i}A'_{\!\!i} \left(P'_{\!\!i}\right)^{-1} = B_{1,i} \bigoplus B_{2,i},\end{equation}

where each $B_{1, i}$ is a Jordan matrix of the form

\begin{equation*}B_{1,i}\;:\!=\;\bigoplus_{j=1}^{s_i} J_{F_{C'_{\!\!i}}^{n_j^{(i)}}, \ell_j^{(i)}},\end{equation*}

where the $\ell_j^{(i)}$ ’s are positive integers and the entries of each $B_{2,i}$ lie inside ${D_i}/{\ell_2}$ for some $\ell_2 \in {\mathbb N}$ . At the expense of replacing $\ell_2$ by a suitable multiple, we may also assume that the entries of each $P_i'$ belong also to ${D_i}/{\ell_2}$ . Moreover, for every $i=1, \dots,r^{\prime}$ the minimal polynomial of $B_{2,i}$ over ${\mathbb Q}(F_{C'_{\!\!i}})$ has no roots that are multiplicatively dependent with respect to $F_{C'_{\!\!i}}$ .

3·12. From linear maps to endomorphisms and finite-to-finite maps

Using the block decomposition given by (3·27), there exists a natural rearrangement of the simple components of $G'_{\!\!2}$ such that:

1. (i) $p\;:\;G'_{\!\!2}\overset{\sim}{\rightarrow} G'_{\!\!3} \times G'_{\!\!4}$ is the isomorphism corresponding to this rearrangement of the simple components of $G'_{\!\!2}$ ;

2. (ii) there exists an endomorphism $\varphi_2$ of $G'_{\!\!3}$ corresponding to $\bigoplus_{i=1}^{r^{\prime}} B_{1,i}$ ; and

3. (iii) there exists a finite-to-finite map $\varphi_3$ on $G'_{\!\!4}$ corresponding to $\bigoplus_{i=1}^{r^{\prime}} B_{2,i}$ .

Let $\lambda\;:\; G'_{\!\!2} \longrightarrow G'_{\!\!2}$ be the endomorphism corresponding to $\left(\ell_2P'_1\right) \oplus \cdots \oplus \left(\ell_2P'_r\right)$ . If we let $g_2 \;:\!=\; \left({\rm id}_{G'_{\!\!1}}, p \circ \lambda \circ \pi'\right)$ , then we obtain the following diagram

(3·28)

Combining (3·28) with (3·18) and (3·6), then we get the following diagram

(3·29)

We note that neither (3·29) nor (3·28) are commutative diagrams since in both cases, the bottom map is only a correspondence (i.e., a finite-to-finite map). On the other hand, both those diagrams are almost commutative, as we will explain next (we refer next to diagram (3·29), but the same argument applies also to diagram (3·28)). So, we let

\begin{equation*}G'\;:\!=\;G'_{\!\!1}\times G'_{\!\!3}\times G'_{\!\!4}\text{ and also, let }\Phi\;:\!=\; (\Phi_1, \varphi_2, \varphi_3)\end{equation*}

and note that there exists $\ell_2\in{\mathbb N}$ such that $[\ell_2]\circ \Phi$ is a well-defined regular morphism of G ^′. Thus, due to our definition of the maps from the diagram (3·29), we get that for each $x\in G$ , we have that

(3·30) \begin{equation}(h\circ\Psi)(x)- (\Phi\circ h)(x)\in G'[\ell_2],\end{equation}

since for any point $y\in G'$ , we have that $\Phi(y)$ consists of finitely many points of the form $z+\xi$ , for some $z\in G'$ and $\xi\in G'[\ell_2]$ .

Furthermore, in light of Remark 3·2, a bit more is true: for any $n\in{\mathbb N}$ , we have that $[\ell_2]\circ \Phi^n$ is a regular self-map on G ^′ and so, for each $x\in G$ , we have that (see also the convention from Remark 3·3)

(3·31) \begin{equation}\left(h\circ\Psi^n\right)(x)- \left(\Phi^n\circ h\right)(x)\in G'[\ell_2].\end{equation}

3·13. The dynamics of finite-to-finite maps

Our goal is to show that in order to prove Theorem 1·3 for the dynamical system $(G,\Psi)$ , it suffices to prove Theorem 1·3 for the dynamical system given by the action of the finite-to-finite map $\Phi$ on $G'=G'_{\!\!1} \times G'_{\!\!3} \times G'_{\!\!4}$ . In order to show this, we first present some general facts regarding the dynamics of the finite-to-finite map $\Phi\;:\;G'\longrightarrow G'$ .

We recall the almost commuting diagram (3·29):

(3·32)

in which case we have that there exists some positive integer $\ell_2$ such that for each $x\in G$ , we have (see also Remark 3·3)

(3·33) \begin{equation}\left(g\circ \Psi - \Phi\circ g\right)(x)\in G'[\ell_2].\end{equation}

Also, very important for our setting is the fact that $G'=G'_{\!\!1}\times G'_{\!\!3}\times G'_{\!\!4}$ and that also $\Phi$ is a split map, i.e., $\Phi=(\Phi_1,\varphi_2,\varphi_3)$ in which $\Phi_1$ is a regular self-map of the semiabelian variety $G'_{\!\!1}$ , and $\varphi_2$ is a group endomorphism of $G'_{\!\!3}$ , while $\varphi_3$ is a finite-to-finite map on $G'_{\!\!4}$ .

In the next Sections we prove that each one of the conclusions (A)-(C) from Theorem 1·3 can be inferred to $(G,\Psi)$ once they are known for $(G',\Phi)$ .

3·14. Condition (a) from Theorem 1·3 transfers from $\Phi$ to $\Psi$

With the notation as in the previous sections (including Definition 3·4), we prove the following result.

Proof. So, we assume there exists a K-point $x\in G'$ with a Zariski dense orbit $\{x_n\}_{n\ge 0}\subset G'(K)$ . We let $y\in G(K)$ such that $g(y)=x$ and we claim that ${\mathcal O}_\Psi(y)$ is Zariski dense in G. Indeed, for each $n\in{\mathbb N}$ , using (3·31), we have that

(3·34) \begin{equation}g\left(\Psi^n(y)\right)-x_n\in G'[\ell_2].\end{equation}

So, letting $\tilde{g}\;:\!=\;[\ell_2]_{G'}\circ g$ be the composition of g with the multiplication-by- $\ell_2$ map on G ^′, we obtain a finite regular map $\tilde{g}\;:\;G\longrightarrow G'$ . Equation (3·34) yields that

(3·35) \begin{equation}\tilde{g}\left(\Psi^n(y)\right)=[\ell_2](x_n)\text{ for each }n\ge 1\end{equation}

and since $\{x_n\}\subset G'$ is Zariski dense, then also the sequence $\{[\ell_2](x_n)\}\subset G'$ is Zariski dense. But then equation (3·35) yields that the orbit ${\mathcal O}_\Psi(y)$ must be Zariski dense in G since $\tilde{g}$ is a finite map.

This concludes our proof of Lemma 3·5.

3·15. Condition (b) from Theorem 1·3 transfers from $\Phi$ to $\Psi$

Lemma 3·6. Assume there exists a non-constant rational function $f\;:\;G'_{\!\!1}\longrightarrow {\mathbb P}^1$ such that

(3·36) \begin{equation}f \circ \Phi_1 = f.\end{equation}

Then there exists a non-constant rational function $f_1\;:\;G\longrightarrow {\mathbb P}^1$ such that $f_1\circ \Psi=f_1$ .

Proof. Let $\Pi\;:\; G'_{\!\!1} \times G'_{\!\!3} \times G'_{\!\!4} \longrightarrow G'_{\!\!1}$ be the projection map onto $G'_{\!\!1}$ . By the diagram (3·29) (see also (3·32)) we must have

(3·37) \begin{equation}\Pi \circ h \circ \Psi = \Phi_1 \circ \Pi \circ h.\end{equation}

Then letting $f_1\;:\!=\;f\circ \Pi \circ h$ (which is still a non-constant rational function since $\Pi \circ h$ is a dominant morphism), we have that

(3·38) \begin{equation}f_1\circ \Psi =f_1,\end{equation}

as desired.

3·16. Condition (c) from Theorem 1·3 transfers from $\Phi$ to $\Psi$

We show that the aforementioned transfer of condition (c) from Theorem 1·3 from the dynamical system $(G',\Phi)$ to the dynamical system $(G,\Psi)$ holds assuming we establish a slightly more precise version of condition (c) in the case $G'=G'_{\!\!1}\times G'_{\!\!3}\times G'_{\!\!4}$ and $\Phi=(\Phi_1,\varphi_2,\varphi_3)$ ; so, with the above notation, we prove the following result.

Lemma 3·8. Let G, G^′ , $\ell_2$ , $\Phi$ , $\Psi$ , g be as in diagram (3·32) and equation (3·33). Assume there exist $n_0\in{\mathbb N}$ , there exists a semiabelian variety Z of dimension larger than ${\rm trdeg}_{\overline{\mathbb{F}_p}}K$ defined over a finite subfield $\mathbb F_q$ of K equipped with the Frobenius endomorphism $F\;:\;Z\longrightarrow Z$ corresponding to $\mathbb F_q$ , and there exists a group homomorphism $\tau\;:\;G'\longrightarrow Z$ such that the following diagram

(3·39)

is almost commuting, i.e., for each $x\in G'(K)$ , we have that (see also Remark 3·3)

(3·40) \begin{equation}\left(\tau\circ \Phi^{n_0}\right)(x) - (F\circ \tau)(x)\in Z[\ell_2].\end{equation}

Then condition (c) of Theorem 1·3 holds for $(G,\Psi)$ .

Remark 3·9. In our proof of Theorem 1·3 for the dynamical system

\begin{equation*}\left(G'_{\!\!1}\times G'_{\!\!3}\times G'_{\!\!4}, \left( \Phi_1, \varphi_2, \varphi_3\right)\right),\end{equation*}

we will show that when condition (c) holds for this dynamical system, then actually there exists a semiabelian variety Z defined over a finite field, along with a dominant group homomorphism $\tau_1\;:\;G'_{\!\!3}\longrightarrow Z$ such that we actually have:

\begin{equation*}\left(\tau_1 \circ \varphi_2^{n_0}\right)(x) - (F\circ \tau_1)(x)\in Z[\ell_2],\end{equation*}

for each $x\in G'_{\!\!3}$ . Then letting $\tau\;:\!=\;\tau_1\circ \Pi$ , where $\Pi$ is the projection of $G'_{\!\!1}\times G'_{\!\!3}\times G'_{\!\!4}$ on the second factor yields the diagram (3·39) and equation (3·40) from Lemma 3·8.

However, for our proof of Lemma 3·8 we do not require the extra information given above that the homomorphism $\tau$ from Lemma 3·8 factors through the projection map $\Pi$ .

Proof of Lemma 3·8. Equation (3·33) and diagram (3·32) yield that for any $x\in G(K)$ , we have

(3·41) \begin{equation}\left(g\circ \Psi^{n_0}\right)(x) - \left(\Phi^{n_0}\circ g\right)(x)\in G'[\ell_2].\end{equation}

Composing with $\tau$ on the left of the equation (3·41) and noting that $\tau\;:\;G'\longrightarrow Z$ is a group homomorphism, we get that for each $x\in G(K)$ , we have

(3·42) \begin{equation}\left(\tau\circ g\circ \Psi^{n_0}\right)(x) - \left(\tau\circ \Phi^{n_0}\circ g\right)(x)\in Z[\ell_2].\end{equation}

On the other hand, equation (3·40) applied to the point $g(x)\in G'(K)$ yields that

(3·43) \begin{equation}\left(\tau\circ \Phi^{n_0}\right)(g(x))-\left(F\circ \tau\right)(g(x))\in Z[\ell_2].\end{equation}

So, combining equations (3·42) and (3·43) yields

(3·44) \begin{equation}\left(\tau\circ g\circ \Psi^{n_0}\right)(x) - \left(F\circ \tau\circ g\right)(x)\in Z[\ell_2].\end{equation}

We let $\tilde{\tau}\;:\!=\;[\ell_2]_Z\circ \tau\circ g$ , which is a dominant group homomorphism $G\longrightarrow Z$ . Equation (3·44) yields that

\begin{equation*}\left(\tilde{\tau}\circ \Psi^{n_0}\right)(x)=\left(F\circ \tilde{\tau}\right)(x),\end{equation*}

for each $x\in G(K)$ , and thus, the following diagram is commutative:

(3·45)

as desired in the conclusion of Lemma 3·8.

3·17. Theorem 1·3 for the simplified dynamical system $(G',\Phi)$

Using Lemmas 3·6, 3·5 and 3·8 we obtain that Theorem 1·3 follows from proving its conclusion for the dynamical system

\begin{equation*}\left(G'_{\!\!1}\times G'_{\!\!3}\times G'_{\!\!4},\left(\Phi_1, \varphi_2, \varphi_3\right)\right),\end{equation*}

as described by the following Theorem.

Theorem 3·10. Let K be an algebraically closed field of positive transcendence degree over $\overline{\mathbb{F}_p}$ , let $G=G_0\times G_1\times G_2$ be a product of reduced split semiabelian varieties defined over $\overline{\mathbb{F}_p}$ , where for each $j=0,1,2$ , we have:

\begin{equation*}G_j\;:\!=\;\prod_{i=1}^{r} C_i^{k_{j,i}}\end{equation*}

for some integers r and $k_{j,i}$ , along with some simple semiabelian varieties $C_i$ such that $C_i$ and $C_{i'}$ are non-isogenous for $i\ne i'$ . Note that we are allowing the integers $k_{i,j}$ to possibly be equal to zero in which case $C_i^{k_{i, j}}$ is equal to the trivial group. We let $\beta\in G_0(K)$ and also let $\varphi_0\in{\rm End}(G_0)$ be a group endomorphism corresponding to a direct sum of unipotent matrices

(3·46) \begin{equation}B_{0,1}\oplus B_{0,2}\oplus \cdots \oplus B_{0,r},\end{equation}

where each $B_{0,i}\in M_{k_{0,i},k_{0,i}}({\rm End}(C_{i}))$ is a direct sum of unipotent Jordan canonical matrices (note that $B_{0,i}$ could possibly be a 0-by-0 matrix, i.e., it may be absent from the direct sum (3·46) if $k_{0,i}=0$ ) of the form

\begin{equation*}J_{1, i_{0,1}^{(j)}} \bigoplus J_{1, i_{0,2}^{(j)} - i_{0,1}^{(j)}} \bigoplus \cdots \bigoplus J_{1, i_{0, \ell_j}^{(j)} - i_{0, \ell_j-1}^{(j)}},\end{equation*}

where $\ell_j \in {\mathbb N}$ and $i_{0,1}, \dots, i_{0, \ell_j}$ are positive integers such that

\begin{equation*}0 < i_{0,1} < i_{0, 2} < \cdots < i_{0, \ell_j} = k_{0, j}.\end{equation*}

We let $\Phi_1\;:\!=\;\tau_{\beta}\circ \varphi_0$ , i.e., the composition of $\varphi_0$ with the translation-by- $\beta$ map on $G_0$ .

We let $\varphi_1\in{\rm End}(G_1)$ be a group endomorphism corresponding to a direct sum of matrices

\begin{equation*}B_{1,1}\oplus B_{1,2}\oplus \cdots \oplus B_{1,r},\end{equation*}

where each $B_{1,j}\in M_{k_{1,j},k_{1,j}}({\rm End}(C_{j}))$ is a direct sum of Jordan canonical matrices of the form:

\begin{equation*}J_{F_{C_{j}}^{n_{1}^{(j)}},i_{1,1}^{(j)}}\oplus J_{F_{C_{j}}^{n_{2}^{(j)}},i_{1,2}^{(j)}-i_{1,1}^{(j)}}\oplus \cdots \oplus J_{F_{C_{j}}^{n_{s_j}^{(j)}},i_{1, s_j}^{(j)}-i_{1, s_j-1}^{(j)}}\end{equation*}

in which $F_{C_{j}}$ is the image in ${\rm End}(C_{j})$ of the Frobenius corresponding to the semiabelian $C_{j}$ , while $s_j, n_1^{(j)},\dots,n_{s_j}^{(j)}\in{\mathbb N}$ and also, $i_{1, 1}^{(j)}, \dots, i_{1, s_j}^{(j)}$ are positive integers such that

\begin{equation*}0 < i_{1,1}^{(j)} < i_{1,2}^{(j)} < \cdots < i_{1, s_j}^{(j)} = k_{1,j},\end{equation*}

for each $j=1,\dots, r$ . (Again, it is possible for $B_{1,j}$ to be a 0-by-0 matrix when $k_{1,j}=0$ .)

We let $\varphi_2\;:\;G_2\longrightarrow G_2$ be a finite-to-finite map corresponding to a direct sum of matrices

\begin{equation*}B_{2,1}\oplus B_{2,2}\oplus \cdots \oplus B_{2,r_2},\end{equation*}

where each $B_{2,i}\in M_{k_{2,i},k_{2,i}}(({1}/{m})\cdot {\rm End}(C_{i}))$ for some given $m\in{\mathbb N}$ . Furthermore, we assume that each matrix $B_{2,i}$ is either a 0-by-0 matrix whenever $k_{2,i}=0$ , or it is NFP (according to Definition 2·6).

We let $\Phi\;:\!=\;\Phi_1\times \varphi_1\times \varphi_2$ be the given correspondence on $G=G_0\times G_1\times G_2$ . Then at least one of the following statements must hold:

1. (a) there exists a point $\alpha\in G(K)$ such that an orbit of $\alpha$ under $\Phi$ (see also the convention from Definition 3·4 regarding the orbit for a finite-to-finite map) is Zariski dense in G;

2. (b) there exists a non-constant rational function $f\;:\;G_0\longrightarrow {\mathbb P}^1$ such that $f\circ \Phi_1=f$ ;

3. (c) there exists a semiabelian variety Z defined over a finite subfield $\mathbb F_q$ of K, endowed with the Frobenius endomorphism $F\;:\;Z\longrightarrow Z$ corresponding to $\mathbb F_q$ , such that

   1. (i) $\dim\!(Z)>{\rm trdeg}_{\overline{\mathbb{F}_p}}K$ ; and

   2. (ii) there exists a dominant group homomorphism $\tau\;:\;G\longrightarrow Z$ and there exist positive integers $\ell_0$ and $n_0$ such that for each $x\in G(K)$ , we have

      \begin{equation*}\left(\tau\circ \Phi^{n_0}\right)(x)-\left(F\circ \tau\right)(x)\in Z[\ell_0],\end{equation*}
      i.e., the following diagram is almost commutative:
      (3·47)

      

The remaining Sections are devoted to proving Theorem 3·10, which in turn yields our main result (Theorem 1·3).