Symbolic dynamics and rotation symmetric Boolean functions

We identify the weights wt(f_n) of a family {f_n} 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 f_n 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(f_n). We apply our results to the case of quadratic functions and considerably extend the results in an earlier paper of ours.