The Orbit Problem consists of determining, given a linear transformation $A$ on $Q^d$, together with vectors $x$ and $y$, whether the orbit of $x$ under repeated applications of $A$ can ever reach $y$.

We will investigate this problem with a different point of view: is it possible to synthesise suitable invariants, that is, subsets of $Q^d$ that contain $x$ but not $y$. Such invariants provide natural certificates for negative instances of the Orbit Problem. We will show that semialgebraic invariants exist in all reasonable cases. A more recent (yet unpublished) result is that existence of semilinear invariants is decidable.

This is a joint work with Nathanaël Fijalkow, Joël Ouaknine, Amaury Pouly and James Worrell, published in STACS 2017.