We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup, the action of on its spectrum is amenable if and only if every action of is amenable.

Ergodic Theory and Dynamical Systems

Exel, R. (Ruy), & Starling, C. (2017). Amenable actions of inverse semigroups. Ergodic Theory and Dynamical Systems, 37(2), 481–489. doi:10.1017/etds.2015.60