@MISC{Dajani_themean, author = {Karma Dajani and Anthony Dooley}, title = {The Mean Ratio Set for Ax+b Valued Cocycles}, year = {} }

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Abstract

Let X = Q 1 i=1 Z `(i) be acted upon by the group \Gamma = \Phi 1 i=1 Z `(i) of changes in finitely many coordinates and ¯ a G-measure on X which is nonsingular for the \Gamma-action on X . We consider cocycles on (X; \Gamma; ¯) 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. 1 Introduction The notion of essential range of real-valued cocycle was defined by Krieger [K] as a subset of [\Gamma1; 1]. He showed that its intersection with (\Gamma1; 1) is a closed subgroup of the real line and that cohomologous cocycles have the same essential range. Parthasarathy and Schmidt [PS] extended this result to cocycles with values in locally compact abelian groups. The notion of essential range has been extended to cocycles with values in general nonabelian locally compact groups, but it i...