www.mdpi.org/entropy On the Measure Entropy of Additive Cellular Automata f∞

We show that for an additive one-dimensional cellular automata f∞ on space of all doubly infinitive sequences with values in a finite set S = {0, 1, 2, ..., r-1}, determined by an additive automaton rule [equation] (mod r), and a f∞-invariant uniform Bernoulli measure μ, the measure-theoretic entropy of the additive one-dimensional cellular automata f∞ with respect to μ is equal to hμ (f∞) = 2klog r, where k ≥ 1, r-1∈S. We also show that the uniform Bernoulli measure is a measure of maximal entropy for additive one-dimensional cellular automata f∞.

on space of all doubly infinitive sequences with values in a finite set S = {0, 1, 2, ..., r-1}, determined by an additive automaton rule f(x ∞ f n-k , ..., x n+k ) = (mod r), and a -invariant uniform Bernoulli measure µ, the measure-theoretic entropy of the additive one-dimensional cellular automata with respect to µ is equal to h where k ≥ 1, r-1∈S.We also show that the uniform Bernoulli measure is a measure of maximal entropy for additive onedimensional cellular automata .

Introduction
Although the additive cellular automata theory and the entropy of this additive cellular automata have grown up somewhat independently, there are strong connections between entropy theory and cellular automata theory.
We give an introduction to additive cellular automata theory and then discuss the entropy of this additive cellular automata.In [1] it was given an algorithm for computing the topological entropy of positively expansive cellular automata.For a definition and some properties of additive onedimensional cellular automata we refer to [2] (see also [1][2][3][4][5][6]).The study of the endomorphisms and the automorphisms (i.e.continuous shift commuting maps, invertible or non-invertible) of the full shift and its subshifts was initialed by Hedlund and coworkers in [3].[4] that if the automaton map is bipermutative then the associated CAaction is strongly mixing and it determines in every direction a Bernoulli transformation with respect to the uniform Bernoulli measure.

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In [6] Ward has used (n, ε)-separated sets to calculate the topological entropy of cellular automata.The definition of additive cellular automata that we have given here differs from the definition given in [6].Detailed information about cellular automata may be found in Wolfram's paper [7].
In order to state our result, we first recall a formulation of our problem.We can also calculate the topological entropy of additive cellular automata .We show that there is a maximal measure for additive cellular automata.

∞ f
The organization of the paper is as follows: In section 2 we establish the basic formulation of problem to state our main Lemma.In section 3 we state our main theorem and prove it.Section 4 contains some remarks.In section 5 there is a conclusion.

Formulation of the Problem and Definitions
Let S ={0, 1, 2, ..., r-1} be a finite alphabet and Ω = S Z be the space of doubly infinite sequence x= , x The shift σ : Ω→ Ω defined by (σx) i = x i+1 is a homeomorphism of compact metric space Ω.Assume that a function f(x -k , ..., x k ) with values in S is given.This function generates a cellular automata of Ω by the formula: is continuous and commutes with left shift (see [3]).
∞ f Definition 1. (see [2]) f is additive if and only if it can be written as where λ i ∈ S. From now on, we will say that a cellular automata is additive if the local rule on which it is based is additive.
Let us consider particular case when f(x n-k , ..., x n+k ) = (mod r).Given integer s ≤ t and a block (i then the partition ξ(-k, k) is a generator for additive cellular automata .
It is evidently ( ) 2 ) ∩ ξ consists of only one cylinder set.Now we calculate the measure of the first preimage of centered cylinder set C = -k [ i -k , ..., i k ] k under additive cellular automata .

∞ f
The Measure Entropy of the Additive One-Dimensional Cellular Automata f ∞ In this section we introduce the measure entropy of the additive cellular automata defined above.Now we can state the main result.
Theorem.Let µ be the uniform Bernoulli measure on Ω with p( i ) = r 1 , for each i =0,1,...,r-1 and f(x n-k , ..., x n+k ) = (mod r).Then measure-theoretic entropy of the additive one-dimensional cellular automata with respect to µ is equal to 2klog r.
Proof.Now we can calculate the measure entropy of the additive cellular automata by using the Kolmogorov-Sinai Theorem [5], namely, and ξ is a generator from Lemma.It is easy to see that if = -r (4k+1) r -(4k+1) log r -(4k+1) = (4k+1)log r.If we continue, then we have the following results: Finally, it is not hard show that H(ξ ∨ ( )

Measures with Maximal Entropy
The metric on Ω is defined by , where d is a metric on S and Ω is a compact metric space.
Definition 2. Let be any uniformly continuous map of a metric space (Ω, ρ).A set E ⊂ Ω is said to be (n, ε)-separated under if for every pair x ≠ y in E there is a ∞ f ∞ f m∈{0, 1, ..., n-1} with the property that ρ( ( ) See [1,6] for the topological entropy of an additive cellular automata .
Definition 3. The measure µ is maximal in the sense that the measure-theoretic entropy of with respect to µ coincides with the topological entropy h

Conclusion
This paper contains the following results: