Katok’s special representation theorem for multidimensional Borel flows

Abstract

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.