Abstract
In the space of binary sequences, minimal sets, that is: sets invariant under the shift operation, that have no invariant proper subsets, are investigated. In applications, such as a piecewise linear circle map and the Smale horseshoe in a mapping of the annulus, each of these sets is invariant under the mapping. These sets can be assigned a unique rotation number equal to the average of the number of ones in the sequences. One special minimal set is the optimal set for which the convergence of the running average to the rotation number is faster than for any other minimal set. These sequences are instrumental in analytically solving for the width of the parameter intervals for which members of a one parameter family of piecewise linear critical circle maps have rational rotation number.
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