arXiv subjects
Publications and source records attributed to Reem Yassawi.
For the action of an algebraic cellular automaton on a Markov subgroup, we show that the Cesàro mean of the iterates of a Markov measure converges to the Haar measure. This is proven by using the combinatorics of the binomial coefficients on the regenerative construction of the Markov measure.
Let M=Z^D be a D-dimensional lattice, and let A be an abelian group. A^M is then a compact abelian group; a `linear cellular automaton' (LCA) is a topological group endomorphism Φ:A^M --> A^M that commutes with all shift maps. Suppose μis a probability measure on A^M whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on Φand μ) under which Φ`asymptotically randomizes' μ, meaning that wk*lim_{J\ni j --> oo} Φ^j μ= η, where ηis the Haar measure on A^M, and J has Cesaro density 1. In the case when Φ=1+σ, we provide a condition on μthat is both necessary and sufficient. We then use this to construct an example of a zero-entropy measure which is asymptotically randomized by 1+σ(all previously known examples had positive entropy).
We consider a left permutive cellular automaton Phi, with no memory and positive anticipation, defined on the space of all doubly infinite sequences with entries from a finite alphabet. For each such automaton that is not one-to-one, there is a dense set of points X (which is large in another sense too) such that the Phi-orbit closure of each x in X is topologically conjugate to an odometer (the ``+1'' map on a projective limit of finite cyclic groups). We identify this odometer in several cases.
This paper has been withdrawn by the authors, due an error involving the weak* convergence argument in section 2
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).
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.