## Abstract

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 C_{0}, C_{1},...,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 M^{2}x =. C_{0} 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 2r^{2} (r^{2} if it is an unstable fp).

Original language | English |
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Pages (from-to) | 145-174 |

Number of pages | 30 |

Journal | Discrete Mathematics |

Volume | 132 |

Issue number | 1-3 |

DOIs | |

State | Published - 15 Sep 1994 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics