Symbolic dynamics and rotation symmetric Boolean functions

Abstract

We identify the weights wt(fn) of a family {fn} of rotation symmetric Boolean functions with the cardinalities of the sets of n-periodic points of a finite-type shift, recovering the second author’s result that said weights satisfy a linear recurrence. Similarly, the weights of idempotent functions fn defined on finite fields can be recovered as the cardinalities of curves over those fields and hence satisfy a linear recurrence as a consequence of the rationality of curves’ zeta functions. Weil’s Riemann hypothesis for curves then provides additional information about wt(fn). We apply our results to the case of quadratic functions and considerably extend the results in an earlier paper of ours.