Rank-two recurrence results for polynomials and questions of dynamical Mordell--Lang type

Let $f,g\in \mathbb{C}[z]\setminus \mathbb{C}$ and $c\in \mathbb{C}[z]$. Suppose that $\mathrm{d}\mathrm{e}\mathrm{g}(c)=1$ if $\mathrm{d}\mathrm{e}\mathrm{g}(f)=\mathrm{d}\mathrm{e}\mathrm{g}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set $$S_{f,g,c}^2:=\{(m,n)\in {\mathbb{Z}}_{\ge 0}^2\text{\hspace{0.17em}⁣}:\mathrm{\exists }\lambda \in \mathbb{C},f^{\circ m}(\lambda )=g^{\circ n}(\lambda )=c(\lambda )\}$$ is semi-linear. This is a generalization of a theorem of Yang and Zhong for the case $m=n$. We also obtain partial results on recurrence sets for rational maps in the case $m=n$. These results are related to higher-dimensional questions of dynamical Mordell–Lang type of rank $\le 2$.