Katok's special representation theorem for multidimensional Borel flows

Abstract

Katok's special representation theorem states that any free ergodic measure-preserving ${\mathbb{R}}^d$-flow can be realized as a special flow over a ${\mathbb{Z}}^d$-action. It provides a multidimensional generalization of the "flow under a function" construction. We prove the analog of Katok's theorem in the framework of Borel dynamics and show that, likewise, all free Borel ${\mathbb{R}}^d$-flows emerge from ${\mathbb{Z}}^d$-actions through the special flow construction using bi-Lipschitz cocycles.