  The persistence behavior for fluctuating steps on the $Si(111)$ $(\sqrt3
\times \sqrt3)R30^{0} - Al$ surface was determined by analyzing time-dependent
STM images for temperatures between 770 and 970K. The measured persistence
probability follows a power law decay with an exponent of $\theta=0.77 \pm
0.03$. This is consistent with the value of $\theta= 3/4$ predicted for
attachment/detachment limited step kinetics. If the persistence analysis is
carried out in terms of return to a fixed reference position, the measured
persistence probability decays exponentially. Numerical studies of the Langevin
equation used to model step motion corroborate the experimental observations.
