Realisation of measured dynamics as uniquely ergodic minimal homeomorphisms on manifolds

TL;DR

This paper demonstrates that measured dynamical systems with certain properties can be realized as uniquely ergodic minimal homeomorphisms on manifolds, and shows stability under extensions, with implications for systems with irrational eigenvalues.

Contribution

It proves stability of realizability as uniquely ergodic minimal homeomorphisms under measured extensions and improves the Jewett-Krieger theorem for ergodic system extensions.

Findings

Measured systems with irrational eigenvalues are isomorphic to minimal homeomorphisms on the two-torus.

Stability of realization under measured extensions on manifolds of dimension at least two.

Enhanced Jewett-Krieger theorem for ergodic system extensions as skew-products on Cantor sets.

Abstract

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss relative version of Jewett-Krieger theorem: any extension between two ergodic systems is isomorphic to a skew-product on Cantor sets.