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In this work, we study arithmetic dynamical questions for formal groups and$p$-adic dynamical systems in higher dimensions. As a generalization ofBerger's result in 1-dimensional case, in the paper, we prove that for any twohigher-dimensional formal groups over the ring of $p$-adic integers, if theyhave infinitely many torsion points in common, then they coincide. Finally, weprove if $\mathcal{D}_1$ and $\mathcal{D}_2$ are $d$-dimensional stable$p$-adic dynamical systems with infinitely many preperiodic points in commonthen $\mathcal{D}_1$ and $\mathcal{D}_2$ coincide.

Unlikely intersection in higher-dimensional formal groups and $p$-adic dynamical systems