Speedups of Ergodic Group Extensions of Z-actions

We define what it means to “speed up” a Zd−measure-preserving dynamical system, and prove that given any ergodic extension Tσ of a Zd− measure-preserving action by a locally compact, second countable group G, and given any second G−extension Sσ of an aperiodic Zd− measure-preserving action, there is a relative speedup of Tσ which is relatively isomorphic to Sσ . Furthermore, we show that given any neighborhood of the identity element of G, the aforementioned speedup can be constructed so that the transfer function associated to the isomorphism between the speedup and Sσ almost surely takes values only in that neighborhood.