The Mean Ratio Set for $ax+b$ Valued Cocycles

Abstract

Let $X={\prod }_{\iota =1}^{\infty }{Z}_{l(\iota )}$ be acted upon by the group $\Gamma ={\oplus }_{\iota =1}^{\infty }{Z}_{l(\iota )}$ of changes in finitely many coordinates and $\mu$ a $G$-measure on $X$ which is nonsingular for the $\Gamma$-action on $X$. We consider cocycles on $(X,\Gamma ,\mu )$ 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.