  Granular convergence is a property of a granular pack as it is repeatedly
sheared in a cyclic, quasistatic fashion, as the packing configuration changes
via discrete events. Under suitable conditions the set of microscopic
configurations encountered converges to a periodic sequence after sufficient
shear cycles. Prior work modeled this evolution as the iteration of a
pre-determined, random map from a set of discrete configurations into itself.
Iterating such a map from a random starting point leads to similar periodic
repetition. This work explores the effect of restricting the randomness of such
maps in order to account for the local nature of the discrete events. The
number of cycles needed for convergence shows similar statistical behavior to
that of numerical granular experiments. The number of cycles in a repeating
period behaves only qualitatively like these granular studies.
