48 research outputs found
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
Lucas congruences for the Ap\'ery numbers modulo
The sequence of Ap\'ery numbers can be interpolated to
by an entire function. We give a formula for the Taylor
coefficients of this function, centered at the origin, as a -linear
combination of multiple zeta values. We then show that for integers whose
base- digits belong to a certain set, satisfies a Lucas congruence
modulo .Comment: 13 pages, 1 figure; significantly shorter proof of Theorem
Semicocycle discontinuities for substitutions and reverse-reading automata
In this article we define the semigroup associated to a substitution. We use
it to construct a minimal automaton which generates a substitution sequence u
in reverse reading. We show, in the case where the substitution has a
coincidence, that this automaton completely describes the semicocycle
discontinuities of u.Comment: 17 pages, 2 figures. This second version is re-written and improve
A family of sand automata
We study some dynamical properties of a family of two-dimensional cellular automata: those that arise from an underlying one-dimensional sand automaton whose local rule is obtained using a Latin square. We identify a simple sand automaton Γ whose local rule is algebraic, and classify this automaton as having equicontinuity points, but not being equicontinuous. We also show that it is not surjective. We generalise some of these results to a wider class of sand automata