  We study the frictional sliding of two ideally incommensurate surfaces with a
third incommensurate sheet - a sort of extended lubricant - in between. When
the mutual ratios of the three periodicities in this sandwich geometry are
chosen to be the golden mean \phi=(1+\sqrt 5)/2, this system is believed to be
statically pinned for any choice of system parameters. In the present study we
overcome this pinning and force the two "substrates" to slide with a mutual
velocity V_ext, analyzing the resulting frictional dynamics. An unexpected
feature is an asymmetry of the relative sliding velocity of the intermediate
lubricating sheet relative to the two substrates. Strikingly, the velocity
asymmetry takes an exactly quantized value which is uniquely determined by the
incommensurability ratio, and absolutely insensitive to all other parameters.
The reason for quantization of the velocity asymmetry will be addressed. This
behavior is compared and contrasted to the corresponding one obtained for a
representative cubic irrational, the spiral mean \omega.
