We prove for $n\geq 3$ that every nonatomic ergodic measure of an $n$-dimensional flow whose Lyapunov exponents off the flow direction are all negative is supported on an attracting periodic orbit.
"On ergodic measures with negative Lyapunov exponents." Proc. Japan Acad. Ser. A Math. Sci. 92 (10) 131 - 135, December 2016. https://doi.org/10.3792/pjaa.92.131