On the Measure Entropy of Additive Cellular Automata f_∞

Abstract

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 f(x_n−k, ..., x_n+k) = ${\displaystyle \sum _{i=-k}^k}{x}_{n+k}$ (mod r), and 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 h_μ ( 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_∞.

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=${(x_n)}_{n=-\infty }^{\infty }$, x_n ∊S. 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 $f_{\infty }$of Ω by the formula:

$$f_{\infty }\text{x}={(y_n)}_{n=-\infty }^{\infty },{\text{y}}_{\text{n}}=f({\text{x}}_{n-k},\dots ,{\text{x}}_{n+k}).$$

Cellular automata $f_{\infty }$ is continuous and commutes with left shift (see [3]).

Definition 1. (see [2]) f is additive if and only if it can be written as

$$f({\text{x}}_{\text{n}-\text{k}},\dots ,{\text{x}}_{\text{n}+\text{k}})={\displaystyle \sum _{i=-k}^k{\lambda }_i{x}_{n+i}}(\text{mod r}),$$

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) = ${\displaystyle \sum _{i=-k}^k{x}_{n+i}}$(mod r). Given integer s ≤ t and a block (i_s ,..., i_t) ∊S^(t-s+1). We define the cylinder set

_s[i_s, ..., i_t ]_t = {x∊ Ω: x_j=i_j, s ≤ j ≤ t }.

Let ξ(s, t) denote the partition of Ω into the cylinder sets of the form _s[i_s, ..., i_t ]_t, where (i_s ,..., i_t) runs over S^t-s+1.

Lemma. Suppose that f(x_n-k, ..., x_n+k)= ${\displaystyle \sum _{i=-k}^k{x}_{n+i}}$(mod r) and ξ(-k, k) is a partition of Ω, where k ≥ 2, then the partition ξ(-k, k) is a generator for additive cellular automata $f_{\infty }$.

Proof. Let ξ(-k, k) ={ _-k[i_-k, ..., i_k ]_k : i_-k, ..., i_k ∊ S} be a partition of Ω. Evidently the partition ξ consist of r^2k+1 elements with same measure r^-(2k+1). If we take the local rule f(x_n-k, ..., x_n+k) = ${\displaystyle \sum _{i=-k}^k{x}_{n+i}}$(mod r), then it is easy to describe ($f_{\infty }$)^-1(ξ). Let us consider a centered cylinder set C = _-k[i_-k, ..., i_k ]_k.

Then it can be writte

( f_∞ )^-1(C) = {{y∊ Ω: y_-2k = j_-2k, ..., y_2k = j_2k}: j_-2k, ..., j_2k ∊S},

where

• y_-2k + ... + y₀ = i_-k (mod r)
• ...
• ...
• y_-k + ... + y_k = i₀ (mod r)
• ....
• ....
• y₀ + ... + y_2k = i_k (mod r).

The partition ξ˅( $f_{\infty }$ )^-1( ξ ) consists of as following elements;

{y∊ Ω: y_-2k = j_-2k, ..., y_2k = j_2k}.

Similarly, it is easy to show that partition ξ ˅ ( $f_{\infty }$ )^-1( ξ )˅ ... ˅ ( $f_{\infty }$ )^-(n-1)( ξ ) consists of as following elements;

{z∊ Ω: z_-nk = i_-nk, ..., z_nk = i_nk },

where i_-nk, ..., i_nk ∊S. Then it is evident that ξ is a generator, that is, $\underset{i=0}{\overset{\infty }{{\displaystyle ˅}}}{(f_{\infty })}^{-i}(\xi )=\epsilon$.

Example. Let S = { 0, 1} and f(x_-2,..., x₂) = ${\displaystyle \sum _{i=-2}^2}$x_i (mod 2). Then we can write

( f∞ )-1(-2[01101] 2)= -4[111100100]4 ∪ -4[111010101]4 ∪ -4[110110110]4

∪-4[101110000]4 ∪ -4[011111100]4 ∪ -4[110000111]4

∪-4[011001101]4 ∪ -4[001101000]4 ∪ -4[000111010]4

∪-4[101000001]4 ∪ -4[010101110]4 ∪ -4[001011001]4

∪-4[010011111]4 ∪ -4[100100010]4 ∪ -4[100010011]4

∪-4[000001011]4.

It is evidently ( f_∞ )^-1 (_-2[01101] 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_∞.

μ(( f_∞ )^-1( C)) = r^2kμ{x∊ Ω: x_-2k = i_-2k, ..., x_2k = i_2k, i_-2k, ..., i_2k∊S}
= r^2kr^-(4k+1) = r^-(2k+1).

Similarly, we can calculate the measure of the second preimage of centered cylinder set C = _-k[ i_-k, ..., i_k]_k under additive cellular automata $f_{\infty }$. It is easy to see that the second preimage of C has amount r^4k as centered cylinder sets

{x∊ Ω: x_-3k = i_-3k, ..., x_3k = i_3k, i_-3k, ..., i_3k ∊S}.

Then we have

μ(( f_∞ )^-2( C)) = r^4kμ{x∊ Ω: x_-3k = i_-3k, ..., x_3k = i_3k, i_-3k, ..., i_3k∊S}
= r^4kr^-(6k+1) = r^-(2k+1).

If we continue this operation, in a similar way, we can determine the measure of the (n-1)st preimages of the centered cylinder set C = _-k[ i_-k, ..., i_k]_k. It is also easy to see that the (n-1)st preimage of the centered block C has amount r^2(n-1)k as centered cylinder sets

{x∊ Ω: x_-nk = i_-nk, ..., x_nk = i_nk, i_-nk, ..., i_nk ∊S}.

We can calculate the measure

μ( f_∞ )^-(n-1)(C)) = r^2(n-1)kμ{x∊Ω: x_-nk = i_-nk, ..., x_nk = i_nk, i_-nk, ..., i_nk∊S}
= r^2(n-1)k r^-(2nk+1) = r^-(2k+1).

It is obvious that the additive one-dimensional cellular automata $f_{\infty }$is a uniform Bernoulli measure-preserving transformation.

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 ) = $\frac{1}{r}$, for each i =0,1,...,r-1 and f(x_n-k, ..., x_n+k) = ${\displaystyle \sum _{i=-k}^k{x}_{n+i}}$ (mod r). Then measure-theoretic entropy of the additive one-dimensional cellular automata $f_{\infty }$ with respect to μ is equal to 2klog r.

Proof. Now we can calculate the measure entropy of the additive cellular automata $f_{\infty }$ by using the Kolmogorov-Sinai Theorem [5], namely,

h_μ ( f_∞ ) = h_μ ( f_∞, ξ ) ,

and ξ is a generator from Lemma. It is easy to see that if

ξ = { _-k[ i_-k, ..., i_k]_k : i_-k, ..., i_k∊S},

then we have

H(ξ) = -r^(2k+1)μ (_-k[ i_-k, ..., i_k]_k )log μ(_-k[ i_-k, ..., i_k]_k )
= -r^(2k+1) r^-(2k+1)log r^-(2k+1)
= (2k+1)log r,

hence

ξ˅ ( f_∞ )^-1(ξ) = {{x∊ Ω: x_-2k = i_-2k, ..., x_2k = i_2k}, i_-2k, ..., i_2k∊S}.

So

H(ξ˅ ( f_∞ )^-1(ξ)) = -r^(4k+1) μ (_-2k[ i_-2k, ..., i_2k]_2k )log μ(_-2k[ i_-2k, ..., i_2k]_2k )
= -r^(4k+1) r^-(4k+1)log r^-(4k+1) = (4k+1)log r.

If we continue, then we have the following results:

ξ ˅ ( f_∞ )^-1(ξ) ˅ ... ˅ ( f_∞ )^-(n-1)(ξ) = {{x∊ Ω: x_-nk = i_-nk, ..., x_nk = i_nk}, i_-nk, ..., i_nk∊S}.

Finally, it is not hard show that

H(ξ ˅ ( f_∞ )^-1(ξ) ˅ ... ˅ ( f_∞ )^-(n-1)(ξ)) = -r^2nk+1 μ (_-nk[ i_-nk, ..., i_nk]_nk )log μ(_-nk[ i_-nk, ..., i_nk]_nk )
= -r^2nk+1 r^-(2nk+1)log r^-(2nk+1) = (2nk+1)log r.

So we have

$${\text{h}}_{\mu }\left(f_{\infty }\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{n}H(\underset{i=0}{\overset{n-1}{˅}}{f}_{\infty }^{-i}(\xi ))=\underset{n\to \infty }{\text{lim}}\frac{2nk+1}{n}\text{log}r=2\text{klog r}.$$

Measures with Maximal Entropy

The metric on Ω is defined by $\rho (x,y)={\displaystyle \sum _{i=-\infty }^{\infty }{2}^{-\left|i\right|}d(x_i,y_i)}$, where d is a metric on S and Ω is a compact metric space.

Definition 2. Let $f_{\infty }$ be any uniformly continuous map of a metric space (Ω, ρ). A set E ⊂ Ω is said to be (n, ε)-separated under $f_{\infty }$ if for every pair x ≠ y in E there is a m∊{0, 1, ..., n-1} with the property that ρ( ( $f_{\infty }$ )^m(x), $f_{\infty }$ )^m(y)) > ε. For each compact set K⊂ Ω, let

s_K (n, ε) = max {|E| : E ⊂ K is (n, ε)-separated under f_∞},

$${\text{h}}_{\text{K}}(f_{\infty },\epsilon )=\underset{n\to \infty }{\text{lim sup}}\frac{1}{n}\text{log}{s}_K(n,\epsilon ),$$

and

$${\text{h}}_{\text{K}}(f_{\infty })=\underset{\epsilon \to 0}{lim}{h}_K(f_{\infty },\epsilon )$$

finally, define the topological entropy of $f_{\infty }$ to be h_top( $f_{\infty }$ ) =$\underset{K}{sup}{h}_K$( $f_{\infty }$ ). See [1,6] for the topological entropy of an additive cellular automata $f_{\infty }$.

Corollary ([6, Corollary]). An additive cellular automata $f_{\infty }$: Ω →Ω with local rule

f (x_-u, ..., x₀, ..., x_k) = ( a_-ux_-u + ... + a₀x₀ + ... + a_kx_k) (mod r),

where a_-u, ..., a₀, ..., a_k∊ S and a_-u, a_k ≠0, has topological entropy

$${\text{h}}_{\text{top}}(f_{\infty })=\{\begin{array}{cc}\text{k log r}& \text{if k}\ge -\text{u}\ge 0,\\ (\text{u}+\text{k})\text{log r}& \text{if k},\text{u}\ge 0,\\ \text{u log r}& \text{if k -u}\le \text{k}\le 0,\end{array}$$

In this paper we assume that a_-u = ... = a₀ = ... = a_k =1.

Definition 3. The measure μ is maximal in the sense that the measure-theoretic entropy of $f_{\infty }$ with respect to μ coincides with the topological entropy h_top( $f_{\infty }$ ).

Remark 1. Let μ be a uniform Bernoulli measure on Ω with p( i ) = $\frac{1}{r}$, for each i=0, 1, ..., r-1, and f(x_n-k, ..., x_n+k) = ${\displaystyle \sum _{i=-k}^k{x}_{n+i}}$ (mod r). Because measure entropy of additive cellular automata $f_{\infty }$with respect to μ is equal to

2klog r = h_μ ( f_∞ ) = h_top( f_∞ ),

the additive cellular automata has a maximal measure.

Remark 2. There are ${(r)}^{r^{2k+1}}$ different functions f :(x_n-k, ..., x_n+k) → S. The function

$$f({\text{x}}_{\text{n}-\text{k}},\dots ,{\text{x}}_{\text{n}+\text{k}})={\displaystyle \sum _{i=-k}^k{x}_{n+i}}(\text{mod r})$$

defines the cellular automata $f_{\infty }$ with maximal entropy among such functions.

Conclusion

This paper contains the following results:

• We have found a generating partition for the additive one-dimensional cellular automata $f_{\infty }$ (Lemma).
• We have calculated the measure-theoretic entropy of the additive one-dimensional cellular automata $f_{\infty }$ (Theorem).
• We have compared the measure-theoretic entropy and the topological entropy of the additive one-dimensional cellular automata. We have seen that the uniform Bernoulli measure is a measure of maximal entropy for additive one-dimensional cellular automata.

In view of Remark 1, an interesting open question is whether there exists a measure of maximal entropy for D-dimensional (D ≥ 2) additive cellular automata over the ring Z_r (r ≥ 2).