Katok's special representation theorem for multidimensional Borel flows

TL;DR

This paper extends Katok's special representation theorem to free Borel -flows, showing they can be constructed from -actions via special flows with bi-Lipschitz cocycles, broadening the theorem's applicability.

Contribution

The paper proves an analog of Katok's theorem in Borel dynamics, demonstrating that all free Borel -flows can be realized as special flows over -actions with bi-Lipschitz cocycles.

Findings

Established a Borel dynamics version of Katok's theorem.

Showed all free Borel -flows arise from -actions.

Extended the 'flow under a function' construction to Borel setting.

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.