On injective endomorphisms of symbolic schemes

Building on the seminal work of Gromov on endomorphisms of symbolic algebraic varieties [10], we introduce a notion of cellular automata over schemes which generalize affine algebraic cellular automata in [7]. We extend known results to this more general setting. We also establish several new ones regarding the closed image property, surjunctivity, reversibility, and invertibility for cellular automata over algebraic varieties with coefficients in an algebraically closed field. As a byproduct, we obtain a negative answer to a question raised in [7] on the existence of a bijective complex affine algebraic cellular automaton $\tau \colon A^{\mathbb Z} \to A^{\mathbb Z}$ whose inverse is not algebraic

On the Ergodic Properties of Certain Additive Cellular Automata over ZmZ_{m}

In this paper, we investigate some ergodic properties of $Z^{2}$-actions $T_{p,n}$ generated by an additive cellular automata and shift acting on the space of all doubly -infinitive sequences taking values in $Z_{m}$.Comment: 5 pag

Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties

We study the dynamical behavior of D-dimensional (D >= 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Giovanni Manzini and Luciano Margara, 1999; Jarkko Kari, 2000; Gianpiero Cattaneo et al., 2000; Gianpiero Cattaneo et al., 2004]. Our main contribution is the proof that topologically transitive additive cellular automata are ergodic. This result represents a solid bridge between the world of measure theory and that of topology theory and greatly extends previous results obtained in [Gianpiero Cattaneo et al., 2000; Giovanni Manzini and Luciano Margara, 1999] for linear CA over Z_m i.e. additive CA in which the alphabet is the cyclic group Z_m and the local rules are linear combinations with coefficients in Z_m. In our scenario, the alphabet is any finite abelian group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over Z_m^n, i.e.with the local rule defined by n x n matrices with elements in Z_m which, in turn, strictly contains the class of linear CA over Z_m. In order to further emphasize that finite abelian groups are more expressive than Z_m we prove that, contrary to what happens in Z_m, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map. As a consequence of our results, we have that, for additive CA, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we have that invertible transitive additive CA are isomorphic to Bernoulli shifts. Finally, we provide a first characterization of strong transitivity for additive CA which we suspect it might be true also for the general case

The algebraic entropy of one-dimensional finitary linear cellular automata

The aim of this paper is to present one-dimensional finitary linear cellular automata $S$ on $\mathbb Z_m$ from an algebraic point of view. Among various other results, we: (i) show that the Pontryagin dual $\widehat S$ of $S$ is a classical one-dimensional linear cellular automaton $T$ on $\mathbb Z_m$; (ii) give several equivalent conditions for $S$ to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of $S$, which coincides with the topological entropy of $T=\widehat S$ by the so-called Bridge Theorem. In order to better understand and describe the entropy we introduce the degree $\mathrm{deg}(S)$ and $\mathrm{deg}(T)$ of $S$ and $T$.Comment: 21 page

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