The r-majority vote action on 0-1 sequences

The r-majority vote operator M replaces simultaneously each digit of a 0-1 sequence x by the majority bit of the cyclic (2r + 1)-interval it centers. This action was suggested and studied as a model for various phenomena in biology [1,2,4], where special attention is given to the set of fixed points (fp's) of this action. We provide an algorithm for M that utilizes the run-size- sequence x̃ of a 0-1 sequence x in order to determine M_x, thereby shedding light on many of M's dynamic properties. We show that for finite x, the number run(x) of x-runs may not increase under M-action, and that it is preserved if and only if x̃∈R^2t is in one of r + 1 convex regions C₀, C₁,...,C_r⊆R. Within C_p, M's action is realized by a circulant matrix containing the row (1, -1, ..., -1, 1, 0, ..., 0) of length 2t with 2p + 1 nonzero entries. An explicit description is obtained for M's fixed points along with all M's regular points, i.e. the x's satisfying M²x =. C₀ contains x̃ iff x is a stable fp, i.e., all x-runs are longer than r. All other regular points lie in the other C_p's and they are all balanced - that is, contain an equal number of zeros and ones - in accordance with Agur's conjecture for finite nonstable fp's [2]. Moreover, any regular sequence that is not a stable fp - be it finite or two-way-infinite - is periodic, with a balanced period not longer than 2r² (r² if it is an unstable fp).