  We demonstrate that stochastic differential equations (SDEs) driven by
fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic
properties as SDEs driven by standard Brownian motion. The focus in this
article is on hypoelliptic systems satisfying H\"ormander's condition. We show
that such systems satisfy a suitable version of the strong Feller property and
we conclude that they admit a unique stationary solution that is physical in
the sense that it does not "look into the future". The main technical result
required for the analysis is a bound on the moments of the inverse of the
Malliavin covariance matrix, conditional on the past of the driving noise.
