Open substacks preserving moduli spaces: classification and bounds

For a smooth locally reductive stack admitting an adequate moduli space, we show that there are finitely many open substacks preserving the adequate moduli space, and give a complete classification of such opens. For ${\mathbb{G}}_m$-actions, this becomes an explicit description in terms of the codimension-one Bialynicki-Birula strata. Moreover, for a stable adequate moduli space, we show that the number of preserving opens are bounded by 3 raised to the relative Picard rank. As an application, the number of fixed components for ${\mathbb{G}}_m$-actions is at least the relative Picard rank.