  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.
