Abstract: The results about the shift map on the limit space of a compact metric space and a sole bonding map is proved: The periodic set of the shift map equals the inverse limit space of the periodic set of the sole bonding map; there is nonrecurrent point in X if there is nonrecurrent point in the inverse limit space; preperiodic point in the inverse limit space must be periodic point; the shift map on the inverse limit space is topologically transitive if its sole bonding map is topologically transitive.