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

### Karma Dajani

Universiteit Utrecht, Netherlands### Anthony H. Dooley

University of Technology Sydney, Ultimo, Australia

## Abstract

Let $X=∏_{ι=1}Z_{l(ι)}$ be acted upon by the group $Γ=⊕_{ι=1}Z_{l(ι)}$ 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.

## Cite this article

Karma Dajani, Anthony H. Dooley, The Mean Ratio Set for $ax+b$ Valued Cocycles. Publ. Res. Inst. Math. Sci. 32 (1996), no. 4, pp. 671–688

DOI 10.2977/PRIMS/1195162716