Reem Yassawi

Limit measures for affine cellular automata II

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

Automatic congruences for diagonals of rational functions

In this paper we use the framework of automatic sequences to study combinatorial sequences modulo prime powers. Given a sequence whose generating function is the diagonal of a rational power series, we provide a method, based on work of Denef and Lipshitz, for computing a nite automaton for the sequence modulo p , for all but nitely many primes p. This method gives completely automatic proofs of known results, establishes a number of new theorems for well-known sequences, and allows us to resolve some conjectures regarding the Ap ery numbers. We also give a second method, which applies to all algebraic sequences, but is signicantly slower. Finally, we show that a broad range of multidimensional sequences possess Lucas products modulo p.

Asymptotic randomization of sofic shifts by linear cellular automata

Let

Orders that yield homeomorphisms on Bratteli diagrams

{\mathbb{M}}={\mathbb{Z}}^D

A characterization of p-automatic sequences as columns of linear cellular automata

be a

Embedding odometers in cellular automata

D

Attractiveness of the Haar measure for linear cellular automata on Markov subgroups

-dimensional lattice, and let

p-adic asymptotic properties of constant-recursive sequences

({\mathcal{A}},+)

A family of sand automata

be an abelian group.

Prevalence of odometers in cellular automata

{\mathcal{A}}^{\mathbb{M}}