Convolution equations on lattices: periodic solutions with values in a prime characteristic field

These notes are inspired by the theory of cellular automata. A linear cellular automaton on a lattice of finite rank or on a toric grid is a discrete dinamical system generated by a convolution operator with kernel concentrated in the nearest neighborhood of the origin. In the present paper we deal with general convolution operators. We propose an approach via harmonic analysis which works over a field of positive characteristic. It occurs that a standard spectral problem for a convolution operator is equivalent to counting points on an associate algebraic hypersurface in a torus according to the torsion orders of their coordinates.Comment: 30 pages, a new editio

Multiband linear cellular automata and endomorphisms of algebraic vector groups

We propose a correspondence between certain multiband linear cellular automata - models of computation widely used in the description of physical phenomena - and endomorphisms of certain algebraic unipotent groups over finite fields. The correspondence is based on the construction of a universal element specialising to a normal generator for any finite field. We use this correspondence to deduce new results concerning the temporal dynamics of such automata, using our prior, purely algebraic, study of the endomorphism ring of vector groups. These produce 'for free' a formula for the number of fixed points of the $n$-iterate in terms of the $p$-adic valuation of $n$, a dichotomy for the Artin-Mazur dynamical zeta function, and an asymptotic formula for the number of periodic orbits. Since multiband linear cellular automata simulate higher order linear automata (in which states depend on finitely many prior temporal states, not just the direct predecessor), the results apply equally well to that class.Comment: 11 page

A short essay on the interplay between algebraic language theory, galois theory and class field theory : comparing physics and theory of computation (Mathematical aspects of quantum fields and related topics)

This paper is written as a technical report for our talk given at the RJMS workshop on quantum fields and related topics, held on 6th- 8th December 2021. In this talk we introduced our recent works [23, 24, 25, 26] in formal language theory to the community of mathematical physics, which concern some interplay between algebraic language theory, galois theory and class field theory. In this paper we discuss some conceptual contents of our recent works [23, 24, 25, 26] in more detail

The Topological Directional Entropy of Z^2-actions Generated by Linear Cellular Automata

In this paper we study the topological and metric directional entropy of $\mathbb{Z}^2$-actions by generated additive cellular automata (CA hereafter), defined by a local rule $f[l, r]$, $l, r\in \mathbb{Z}$, $l\leq r$, i.e. the maps $T_{f[l, r]}: \mathbb{Z}^\mathbb{Z}_{m} \to \mathbb{Z}^\mathbb{Z}_{m}$ which are given by $T_{f[l, r]}(x) =(y_n)_ {-\infty}^{\infty}$, $y_{n} = f(x_{n+l}, ..., x_{n+r}) = \sum_{i=l}^r\lambda_{i}x_{i+n}(mod m)$, $x=(x_n)_ {n=-\infty}^{\infty}\in \mathbb{Z}^\mathbb{Z}_{m}$, and $f: \mathbb{Z}_{m}^{r-l+1}\to \mathbb{Z}_{m}$, over the ring $\mathbb{Z}_m (m \geq 2)$, and the shift map acting on compact metric space $\mathbb{Z}^\mathbb{Z}_{m}$, where $m$ $(m \geq2)$ is a positive integer. Our main aim is to give an algorithm for computing the topological directional entropy of the $\mathbb{Z}^2$-actions generated by the additive CA and the shift map. Thus, we ask to give a closed formula for the topological directional entropy of $\mathbb{Z}^2$-action generated by the pair $(T_{f[l, r]}, \sigma)$ in the direction $\theta$ that can be efficiently and rightly computed by means of the coefficients of the local rule f as similar to [Theor. Comput. Sci. 290 (2003) 1629-1646]. We generalize the results obtained by Ak\i n [The topological entropy of invertible cellular automata, J. Comput. Appl. Math. 213 (2) (2008) 501-508] to the topological entropy of any invertible linear CA.Comment: 9 pages. submitte