Measures and stabilizers of group Cantor actions

We consider a minimal equicontinuous action of finitely generated group $ G on Cantor set X with invariant probability measure \mu $, and the stabilizers points for such an action. give sufficient conditions under which there exists subgroup H that in whose are conjugate to has full measure. The is locally quasi-analytic non-degenerate. An if its elements have unique extensions subsets uniform diameter. condition non-degenerate introduced this paper. apply our results study properties random subgroups induced by actions sets certain almost one-to-one actions.