The Mean Ratio Set for <i>ax</i> + <i>b</i> Valued Cocycles

Abstract

Let X=Π∞_t_=1_Zl_(i) be acted upon by the group Γ=⊕∞_i_=1_Zl_(i) of changes in finitely many coordinates and μ a G-measure on X which is nonsingular for the Γ-action on X. We consider cocycles on (X, Γ, μ) taking values in the ax + b group. We give a structure theorem for such cocycles, we define the mean ratio set which is a closed subgroup of the ax + b group and we exhibit for each closed subgroup a cocycle whose mean ratio set is the given subgroup.