arXiv subjects

Marcus Pivato

Publications and source records attributed to Marcus Pivato.

22 records · Page 2Linked to original sources

arXiv2002

Limit Measures for Affine Cellular Automata

Let M be a monoid (e.g. the lattice Z^D), and A an abelian group. A^M is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F:A^M --> A^M that commutes with all shift maps. Let mu be a (possibly nonstationary) probability measure on A^M; we develop sufficient conditions on mu and F so that the sequence {F^N mu} (N=1,2,3,...) weak*-converges to the Haar measure on A^M, in density (and thus, in Cesaro average as well). As an application, we show: if A=Z/p (p prime), F is any ``nontrivial'' LCA on A^{(Z^D)}, and mu belongs to a broad class of measures (including most Bernoulli measures (for D >= 1) and ``fully supported'' N-step Markov measures (when D=1), then F^N mu weak*-converges to Haar measure in density.

math.DS↗

arXiv2002

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

If M is a monoid (e.g. the lattice Z^D), and G is a finite (nonabelian) group, then G^M is a compact group; a `multiplicative cellular automaton' (MCA) is a continuous transformation F:G^M-->G^M which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of G. We characterize when MCA are group endomorphisms of G^M, and show that MCA on G^M inherit a natural structure theory from the structure of G. We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.

math.DS↗

arXiv2002

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.

math.DS↗

arXiv2001

Building a Stationary Stochastic Process From a Finite-dimensional Marginal

If A is a finite alphabet, Z^D is a D-dimensional lattice, U is a subset of Z^D, and mu_U is a probability measure on A^U that ``looks like'' the marginal projection of a stationary random field on A^(Z^D), then can we ``extend'' mu_U to such a field? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when D = 1, we provide some sufficient conditions and some necessary conditions for mu_U to be extendible for D > 1, and show that, in general, the problem is not formally decidable.

math.PR↗