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 “On the Greenfield-Wallach and Katok conjectures in dimension three,” Contemporary Mathematics 469 (2008).
"On manifolds supporting distributionally uniquely ergodic diffeomorphisms." J. Differential Geom. 99 (2) 191 - 213, February 2015. https://doi.org/10.4310/jdg/1421415561