Let \(K\) be an algebraic number field, \(f \in K(X)\) a rational function of degree \(d \ge 2\) and \(\Gamma\) a finitely generated subgroup of the multiplicative group \(K^\ast\). For \(\alpha \in K\) let \(\mathcal O_f (\alpha) = \{f^{(n)}(\alpha) \mid n \ge 0\}\) denote the orbit of \(\alpha\) under \(f\), where \(f^{(n)}\) denotes the \(n\)-th iterate of \(f\).
The authors investigate whether elements of \(\mathcal O_f (\alpha)\) are multiplicatively dependent modulo \(\Gamma\), more precisely, they study the tuples of solutions \((n,k,\alpha,r,s)\) to the relation \[ f^{(n+k)}(\alpha)^r \cdot f^{(k)}(\alpha)^s \in \Gamma\,,\tag{1} \] where \(\alpha \in K\) and \(k,n,r,s \in \mathbb Z\) with \(0\le k\), \(1\le n\) and \((r,s) \ne (0,0)\). Studying finiteness results, one obviously can restrict to those \(\alpha\) with infinite orbit, and also replace \(\Gamma\) by a group of \(S\)-units for some finite set \(S\) of valuations of \(K\).
Theorem 1.2 characterizes all rational functions \(f\) for which the following can happen: there are infinitely many \(\alpha \in K\) with \(f(\alpha) \in \Gamma\), or there are infinitely many \((n,\alpha) \in \mathbb Z_{\ge 2} \times K\) with infinite orbit \(\mathcal O_f (\alpha)\) and \(f^{(n)} (\alpha) \in \Gamma\). The authors give a new short proof for this known result. They use Lemma 2.7, which calculates the genus of curves which generalize superelliptic curves, and Falting’s Theorem.
Supposing that \(0\) is not a periodic point of \(f\), it is shown in Theorem 1.3 that for fixed integers \(r,s\) with \(rs \ne 0\) if (1) has infinitely solutions with \(n\) exceeding some given bound and with infinite orbit \(\mathcal O_f (\alpha)\) then the number of zeroes and poles of \(f\) is at most \(2\). Beside Theorem 1.2, a key ingredient for the proof is Lemma 2.5, which is a dynamical Diophantine approximation result of L.-C. Hsia and J.-H. Silverman [Pac. J. Math. 249, No. 2, 321–342 (2011; Zbl 1272.37039)] and which relies on a suitable version of Roth’s Theorem.
Finally, Theorem 1.4 lists all rational functions \(f\) for which it can happen that (1) has infinitely many solutions with infinite orbit \(\mathcal O_f (\alpha)\) for \(r=s=1\) and \(n \ge 1\), or \(n \ge 2\), resp. The proof uses the preceeding theorems and again Lemma 2.7, and leads to a detailed case-by-case study.
Specializing to the case that \(f\) is a polynomial, Theorem 1.7 shows that under some mild restrictions on \(f\) relation (1) has always finitely many solutions. Besides the preceding results, the proof uses Lemma 2.8, which is a general version of the theorem of A. Schinzel and R. Tijdeman [Acta Arith. 31, 199–204 (1976; Zbl 0339.10018)].
In the concluding Theorem 1.10, multiplicative dependence of the elements of \(\mathcal O_f (\alpha)\) is generalized by “multilinear polynomials with split variables”.