Invariant measures for bipermutative cellular automata (original) (raw)

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

2019

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 alphab...

Right-permutative cellular automata on topological Markov chains

Discrete and Continuous Dynamical Systems, 2008

In this paper we consider cellular automata (G, Φ) with algebraic local rules and such that G is a topological Markov chain which has a structure compatible to this local rule. We characterize such cellular automata and study the convergence of the Cesàro mean distribution of the iterates of any probability measure with complete connections and summable decay.

Some ergodic properties of invertible cellular automata

In this paper we consider invertible one-dimensional linear cellular automata (CA hereafter) defined on a finite alphabet of cardinality p k , i.e. the maps T f [l,r] : Z Z p k → Z Z p k which are given by T f [l,r] (x) = (yn) ∞ n=−∞ , yn = f (x n+l ,. .. , x n+r) = r P i=l λ i x n+i (mod p k), x = (xn) ∞ n=−∞ ∈ Z Z p k and f : Z r−l+1 p k → Z p k , over the ring Z p k (k ≥ 2 and p is a prime number), where gcd(p, λr) = 1 and p|λ i for all i = r (or gcd(p, λ l) = 1 and p|λ i for all i = l). Under some assumptions we prove that any right (left) permutative, invertible one-dimensional linear CA T f [l,r] and its inverse are strong mixing. We also prove that any right(left) permutative, invertible one-dimensional linear CA is Bernoulli automorphism without making use of the natural extension previously used in the literature.

Remark on cellular automata and shift preserving maps

Applied Mathematics Letters, 2006

The main goal of this work is to show an extension of well known Hedlund's theorem which states that in the Cantor topology the cellular automata are the continuous shift preserving maps. This extension maintains the topological structure and uses the concept of barriers of Set Theory to generalize the notion of local rule in the definition of cellular automata.

Conservation of some dynamical properties for operations on cellular automata

Theoretical Computer Science, 2009

We consider the family of all the Cellular Automata (CA) sharing the same local rule but having different memory. This family contains also all the CA with memory m ≤ 0 (one-sided CA) which can act both on A Z and on A N . We study several set theoretical and topological properties for these classes. In particular, we investigate if the properties of a given CA are preserved when considering the CA obtained by changing the memory of the original one (shifting operation). Furthermore, we focus our attention to the one-sided CA acting on A Z starting from the one-sided CA acting on A N and having the same local rule (lifting operation). As a particular consequence of these investigations, we prove that the long-standing conjecture [Surjectivity ⇒ Dense Periodic Orbits (DPO)] can be restated in several different (but equivalent) ways. Furthermore, we give some results on properties conserved under iteration of the CA global map.

On strong mixing property of cellular automata with respect to Markov measures

In this paper we study mixing properties of one-dimensional linear cellular automata over the ring Z m , a particular class of dynamical systems, determined by right (left) permutative local rule F with respect to the uniform Markov measure induced by doubly stochastic matrix P = p (i,j) and the probability vector π. We prove that one-dimensional linear cellular automata associated to right (resp. left) permutative local rule F = F [l, r], 0 < l < r (resp. l < r < 0), is strong mixing with respect to the uniform Markov measure. We also show that Z × N-actions generated by the one-dimensional linear cellular automata determined by bipermutative local rule F = F [l, r] and shift map is strong mixing with respect to the uniform Markov measure.

A non-ergodic probabilistic cellular automaton with a unique invariant measure

Stochastic Processes and their Applications, 2011

We exhibit a Probabilistic Cellular Automaton (PCA) on {0, 1} Z with a neighborhood of size 2 which is non-ergodic although it has a unique invariant measure. This answers by the negative an old open question on whether uniqueness of the invariant measure implies ergodicity for a PCA.