On manifolds supporting distributionally uniquely ergodic diffeomorphisms

TL;DR

This paper constructs new examples of distributionally uniquely ergodic diffeomorphisms on manifolds other than tori, challenging a previous conjecture and expanding understanding of ergodic theory on manifolds.

Contribution

It provides the first known examples of DUE diffeomorphisms on non-torus manifolds, countering Forni's conjecture about their support.

Findings

Constructed DUE diffeomorphisms on closed manifolds other than tori

Counterexamples to Forni's conjecture

Expanded the class of manifolds known to support DUE diffeomorphisms

Abstract

A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this article we construct DUE diffeomorphisms supported on closed manifolds different from tori, providing some counterexamples to a conjecture proposed by Forni in [For08].