Right-Permutative Cellular Automata on Topological Markov Chains

In this paper we consider cellular automata $(\mathfrak{G},\Phi)$ with algebraic local rules and such that $\mathfrak{G}$ is a topological Markov chain which has a structure compatible to this local rule. We characterize such cellular automata and study the convergence of the Ces\`aro mean distribution of the iterates of any probability measure with complete connections and summable decay.Comment: 16 pages, 2 figure. A new version with improved redaction of Theorem 6.3(i)) to clearify its consequence

Conservation Laws in Cellular Automata

If X is a discrete abelian group and B a finite set, then a cellular automaton (CA) is a continuous map F:B^X-->B^X that commutes with all X-shifts. If g is a real-valued function on B, then, for any b in B^X, we define G(b) to be the sum over all x in X of g(b_x) (if finite). We say g is `conserved' by F if G is constant under the action of F. We characterize such `conservation laws' in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all one-dimensional CA exhibiting a given conservation law.Comment: 19 pages, LaTeX 2E with one (1) Encapsulated PostScript figure. To appear in Nonlinearity. (v2) minor changes/corrections; new references added to bibliograph

Symmetry groupoids and patterns of synchrony in coupled cell networks

A coupled cell system is a network of dynamical systems, or “cells,” coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only mechanism that can create such states in a coupled cell system and show that it is not. The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph—the dynamical systems with the corresponding internal dynamics and couplings—are precisely those that are equivariant under the symmetry groupoid. A pattern of synchrony is “robust” if it arises for all admissible vector fields. The first main result shows that robust patterns of synchrony (invariance of “polydiagonal” subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an equivalence relation on cells is “balanced.” The second main result shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled cell network, the “quotient network.” The existence of quotient networks has surprising implications for synchronous dynamics in coupled cell systems

Risky social choice with approximate interpersonal comparisons of well-being

We develop a model of social choice over lotteries, where people's psychological characteristics are mutable, their preferences may be incomplete, and approximate interpersonal comparisons of well-being are possible. Formally, we suppose individual preferences are described by a von~Neumann-Morgenstern (vNM) preference order on a space of lotteries over psychophysical states; the social planner must construct a vNM preference order on lotteries over social states. First we consider a model when the individual vNM preference order is incomplete (so not all interpersonal comparisons are possible). Then we consider a model where the individual vNM preference order is complete, but unknown to the planner, and thus modeled by a random variable. In both cases, we obtain characterizations of a utilitarian social welfare function

Limit Measures for Affine Cellular Automata, II

If M is a monoid (e.g. the lattice Z^D), and A is an abelian group, then A^M is a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F:A^M --> A^M that commutes with all shift maps. If F is diffusive, and mu is a harmonically mixing (HM) probability measure on A^M, then the sequence {F^N mu} (N=1,2,3,...) weak*-converges to the Haar measure on A^M, in density. Fully supported Markov measures on A^Z are HM, and nontrivial LCA on A^{Z^D} are diffusive when A=Z/p is a prime cyclic group. In the present work, we provide sufficient conditions for diffusion of LCA on A^{Z^D} when A=Z/n is any cyclic group or when A=[Z/(p^r)]^J (p prime). We show that any fully supported Markov random field on A^{Z^D} is HM (where A is any abelian group).Comment: LaTeX2E Format, 20 pages, 1 LaTeX figure, 2 EPS figures, to appear in Ergodic Theory and Dynamical Systems, submitted April 200