Polynomial dynamics and local analysis of small and grand orbits

We prove an analogue of the Manin-Mumford conjecture for polynomial dynamical systems over number fields. In our setting the role of torsion points is taken by the small orbit of a point $\alpha$. The small orbit of a point was introduced by McMullen and Sullivan in their study of the dynamics of rational maps where for a point $\alpha$ and a polynomial $f$ it is given by \begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*} Our main theorem is a classification of the algebraic relations that hold between infinitely pairs of points in $\mathcal{S}_\alpha$ when everything is defined over the algebraic numbers and the degree $d$ of $f$ is at least 2. Our proof relies on a careful study of localizations of the dynamical system and follows an entirely different approach than previous proofs in this area. At infinite places of $K$ we use known rigidity theorems of Fatou and Levin to prove new such. These might be of independent interest in complex dynamics. At finite places we introduce new non-archimedean methods to study diophantine problems that might be applicable in other arithmetic contexts. Our method at finite places allows us to classify all algebraic relations that hold for infinitely pairs of points in the grand orbit \begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*} of $\alpha$ if $|f^{\circ n}(\alpha)|_v \rightarrow \infty$ at a finite place $v$ of good reduction co-prime to $d$ . This is an analogue of the Mordell-Lang conjecture on finite rank groups for polynomial dynamics.