Open ${\mathbb{C}\mathbb{P}}^1$ descendent theory I: The stationary sector

We define stationary descendent integrals on the moduli space of stable maps from disks to $({\mathbb{C}\mathbb{P}}^1,{\mathbb{R}\mathbb{P}}^1)$. We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus $0$ disk cover invariants in the stationary case. For all higher genus invariants, we propose a conjectural formula.