The Complexity of Permutive Cellular Automata

The complexity of permutive cellular automata

a_{Department of Applied Mathematics, National Dong Hwa University, Hualian 97063, Taiwan,}

R.O.C.

b_{Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan,}

R.O.C.

c_{Department of Applied Mathematics, National Pingtung University of Education, Pingtung 90003,}

Taiwan, R.O.C.

Abstract

This paper studies cellular automata in two aspects: Ergodic and topological behavior. For ergodic aspect, the formulae of measure-theoretic entropy, topological entropy and topological pressure are given in closed forms and Parry measure is demonstrated to be an equilibrium measure for some potential function. For topological aspect, an example is examined to show that the exhibition of snap-back repellers for a cellular automaton infers Li-Yorke chaos. In addition, bipermutive cellular automata are optimized for the exhibition of snap-back repellers in permutive cellular automata whenever two-sided shift space is considered.

Key words: Cellular automata, permutive, equilibrium measure, Parry measure, snap-back repeller, chaos

2000 MSC: 28D20, 37B15, 37B40, 47A35

1. Introduction

Cellular automaton (CA), introduced by Ulam [1] and von Neumann [2] as a model for self-production, is a particular class of dynamical systems which is defined by a local rule acting on a discrete space and is widely studied in a variety of contexts in physics, biology and computer science.

CA is brought to the attention of wide audience through an ecological model, say Conway’s Game-of-Life, which consists of simple rule but leads to complex behavior. Physicists use CA to investigate many phenomena such as spiral waves, phase transi-tions, reaction-diffusion processes, fluid, and so on. In 1980s, Hardy, Pazzis and Pomeau introduced the so-called HPP lattice gas model to study fundamental statistical prop-erties of a gas of interacting particles. Greenberg and Hastings yield CA to investi-gate reaction-diffusion equations and find that such a simplified model can still gen-erate spatial-temporal structures similar to the original system. Reader is referred to [4, 5, 6, 7, 8, 9, 10, 3, 11, 12, 13, 14, 15] and references therein for more details.

Email addresses: [email protected] (Jung-Chao Ban), [email protected]

(Chih-Hung Chang), [email protected] (Ting-Ju Chen), [email protected] (Mei-Shao Lin)

One-dimensional CA consists of infinite lattice with finite states and a local rule. Hedlund [16] discusses CA systematically from purely mathematical point of view. Wol-fram [17, 18] also makes a decisive impulse to the mathematical study; he proposes a classification of CA by means of asymptotical dynamics. After the topological behavior of CA were widely elucidated [19, 20, 21], the decidability of topological entropy causes researcher’s interest. Hurd et al. [22] indicate that there exists no algorithm for the computation of topological entropy in general but case by case. Recently, a closed form of formula for topological entropy of additive CA is demonstrated [23, 24].

This elucidation concerns phenomena presented by permutive CA that is initiated by Hedlund. Permutive CA is widely investigated and asserts rich properties [16, 25]. This study is divided into two aspects: The viewpoints of ergodic theory and topological behavior.

In the first part, the ergodic properties such as measure-theoretic entropy, topological entropy and topological pressure are elucidated. Let F be an additive CA, i.e., the local rule of F is a linear function, which is associated with prime states, Ward [24] gives a closed formula for the calculation of topological entropy. This investigation extends Ward’s formula to permutive CA. Notably, permutive CA need not be additive and additive CA with prime states is a proper subset of permutive CA. D’amico et al. [23] demonstrate an algorithm for the computation of topological entropy for expansive CA, our result gives topological entropy a compact form for a subset of expansive CA and also works for some permutive but non-expansive cases.

Lind [26] gives an application of ergodic theory in CA and shows that the Ces`aro mean of any Bernoulli measure iterated by CA with nearest neighborhood in two-symbolic shift space converges to the uniform measure. Sablik [27] studies measure-theoretic entropy of bipermutive CA and indicates that the uniform Bernoulli measure is a unique invariant measure for some invertible affine expansive CA. A classical variational principle for thermodynamics says that, if X is compact and T : X → X is continuous, the supremum of measure-theoretic entropy coincides with topological entropy, where the supremum is taken over all ergodic measures. An ergodic measure that attains the supremum is called a measure with maximum entropy and the uniform Bernoulli measure is demonstrated to be one for permutive CA. Moreover, topological entropy can be extended to the consideration of topological pressure whenever a potential function φ∈ C(X, R) is given and topological entropy is a special case for topological pressure with null potential. Also, variational principle for topological pressure asserts that topological pressure is equal to the supremum of the sum of measure-theoretic entropy and integration of potential function. The formulae of measure-theoretic entropy and topological pressure infer that Parry measure is an equilibrium measure for some potential function; herein an invariant measure is called an equilibrium measure provided it reaches the supremum.

The second part studies the complexity of the topological behavior exhibited by per-mutive CA. Li and Yorke [28] discover that, if a first-order difference equation

x_i+1= f (x_i), i∈ Z+,

where x_i ∈ R and f : R → R is continuous, admits a 3-cycle, then there exist many complex behavior: The lack of global stability and the existence of an uncountable set of orbits which do not approach any periodic path. Those systems assert such phenomena are called Li-Yorke chaos hereafter. Marotto [29, 30] demonstrates the existence of a

snap-back repeller implements Li-Yorke chaos. Garc´ıa [31] also shows the existence of snap-back repeller admits positive topological entropy, which is a sufficient condition for Li-Yorke chaos [32].

This is a motivation that, in CA, the existence of snap-back repeller also implements Li-Yorke chaos. Notably, the dynamical system considered in [29] is differentiable. CA, however, is only a continuous system. A definition of snap-back repeller for discrete dynamical systems is given and, instead of serious proof, an example is studied to convince that the existence of snap-back repeller implement Li-Yorke chaos. Furthermore, each bipermutive CA exhibits a snap-back repeller.

The rest of this elucidation is organized as follows. Section 2 gives some preliminaries. Section 3 studies the ergodic properties of permutive CA while Section 4 investigates the existence of snap-back repeller implies Li-Yorke chaos and bipermutive CA is a collection which exhibits snap-back repeller. Some conclusions are given in Section 5.

2. Notation and Definition

Let S ={0, 1, 2, . . . , r − 1} be a finite alphabet and let Ω = SZ be the space of bi-infinite sequence x = (x_n)∞_−∞. Hedlund [16] examines CA in the viewpoint of symbolic dynamics.

Theorem 2.1 ([16]). A map F : Ω→ Ω is a CA if and only if F can be represented as a sliding block code, i.e., there exists k∈ Z+ and a block map f : S2k+1→ S such that

F (x)_i= f (x_i−k, . . . , x_i+k)

for x∈ Ω and i ∈ Z.

Such f is called the local rule of F . The study of the local rule of a CA is essential for the understanding of this system. A class of local rules, say permutive, is initially introduced by Hedlund [16].

Definition 2.2. The local rule f : S2k+1 → S for a given CA is said to be leftmost (respectively rightmost) permutive if there exists an integer i,−k ≤ i ≤ −1 (respectively

1≤ i ≤ k), such that

(i) f is a permutation at x_i whenever the other variables are fixed;

(ii) f does not depend on x_j for j < i (respectively j > i).

Example 2.3. Let S ={0, 1} be the alphabet and let f : S3→ S be a local rule defined by

f (x₋₁, x₀, x₁) = x₀+ x₁ mod 2.

For any pair (a₁, a₂)∈ S2, the transformation g(x)≡ f(a₁, a₂, x) : S→ S is a

permuta-tion. Thus f is rightmost permutive.

The local rule f : S2k+1→ S is called bipermutive if f is both leftmost and rightmost permutive. In the rest of this work, f is called permutive provided f is either one of the following three cases:

(i) f is leftmost permutive and does not depend on x_i for i > 0; 3

(ii) f is rightmost permutive and does not depend on x_i for i < 0; (iii) f is bipermutive.

Hedlund demonstrates a permutive CA is surjective.

Proposition 2.4 ([16]). If f is permutive, then F is surjective.

Define d : Ω× Ω → R by d(x, y) = ∞  i=−∞ |xi− yi| r|i| , x, y∈ Ω. (1)

It is easy to verify that d is a metric and (Ω, d) is a compact metric space. Moreover, let

a[sa, . . . , sb]b ={x ∈ Ω : xa= sa, . . . , xb = sb} be a cylinder in Ω, where a ≤ b, a, b ∈ Z.

Then_a[s_a, . . . , s_b]_b is not only open but close in Ω.

Consider μ an invariant probability measure on (Ω, F ). Let α and β be two finite measurable partitions of Ω, define

αβ ={AB : A∈ α, B ∈ β}.

It is easily seen that αβ is a refinement of α and β. Denote by H_μ(α) =−

A∈α

μ(A) log μ(A).

The measure-theoretic entropy of α is defined by

h_μ(F, α) = lim n→∞ 1 nHμ( n−1_ m=0 F−mα), (2)

reader may refer to [33] for the existence of the limit. And the measure-theoretic entropy of F is defined by

h_μ(F ) = sup h_μ(F, α), (3)

where the supremum is taken over all finite measurable partitions of Ω. LetP be an open cover of Ω, denote by

H(P) = inf{log card ˆP},

where the infimum is taken over the set of finite subcovers ˆP of P and card A is the cardinality of A. The topological entropy ofP is defined by

h_top(F,P) = lim n→∞ 1 nH( n−1_ m=0 F−mP). (4)

For the existence of the limit, reader is referred to [33] for more details. The topological entropy of F is defined by

h_top(F ) = sup h_top(F,P), (5)

where the supremum is taken over all open covers of Ω. 4

In addition, for α an open cover of Ω and φ∈ C(Ω, R) a continuous function from Ω to R, denote by p_n(F, φ, α) = inf{ B∈β sup x∈Be (Snφ)(x)_{: β is a finite subcover of} n−1_ m=0 F−mα},

where n∈ N and S_nφ =n−1_m=0φ◦ Fm. Then lim_n→∞n1log p_n(F, φ, α) exists [33]. For each δ > 0, define P (F, φ, δ) = sup{ lim n→∞ 1 nlog pn(F, φ, α) : diam(α)≤ δ}, (6) and P (F, φ) = lim δ→0P (F, φ, δ). (7)

The map P (F,·) : C(Ω, R) → R ∪ {∞} is called the topological pressure of F . It comes immediately that P (F, 0) = h_top(F ).

3. Ergodic properties of permutive cellular automata

This section investigates ergodic properties of permutive CA. First assume that f is leftmost permutive at x_−i (0 < i≤ k). Set j = max{ : f depends on x_}, then j ≤ 0 and f can be expressed in a compact form f = f (x_−i, . . . , x_j) : Si+j+1→ S.

3.1. Measure-theoretic entropy

LetB be a Borel σ-algebra on Ω, μ = (p₀, p₁, . . . , p_r−1) be an F -invariant Bernoulli measure, i.e.,

μ(_a[s_a, . . . , s_b]_b) = p_sa· · · p_sb, for_a[s_a, . . . , s_b]_b ⊂ Ω.

For ∈ N, denote by ξ_ ={_−[s_−, . . . , s_]_ : s_−, . . . , s_ ∈ S} a measurable partition of

Ω.

Lemma 3.1. Let ξ_ be a partition of Ω for some  ∈ N, then μ(n−1_m=0F−mξ_) → 0

as n → ∞, where μ(α) ≡ sup{μ(A) : A ∈ α} and α is a partition of Ω. Moreover,

_n−1

m=0F−mξ= ξ(−−(n−1)i, ) provided  large enough, where ξ(a, b) = {a[xa, . . . , xb]b:

x_a, . . . , x_b∈ S}.

Proof. First observe that for each z = (za, . . . , zb)∈ Sb−a+1, fb−a−i−j−1 z∈ Sb−a+i+j+1,

where f_m: Si+j+m+1→ Smis defined by

f_m(x_−i−m, . . . , x_j) = (f (x_−i−m, . . . , x_j−m), . . . , f (x_−i, . . . , x_j)), for m∈ N, and f₀ = f . Denote f_{b−a−i−j} by f without ambiguity. For s_b−i+1, s_b−i+2, . . . , s_b+j ∈ S, the leftmost permutivity of f at x_−i implies there admits a unique ˜z_b−i such that

f (˜z_b−i, s_b−i+1, . . . , s_b+j) = z_b. Repeating this process there are uniquely determined ˜

z_b−i−1, . . . , ˜z_a−i∈ S such that

f (˜z) = z, where ˜z = (˜z_a−i, . . . , ˜z_b−i, s_b−i+1, . . . , s_b+j)∈ Sb−a+i+j+1.

Denote by ξ_={A_l}r_l=12+1, above discussion asserts that

F−1A_n1F−1A_n2=∅, for n₁= n₂.

More than that, it is easily seen that F−mA_n1F−mA_n2 =∅ for n₁= n₂, m∈ Z+. For each B ∈ ξ_F−1ξ_, there exists n₁, n₂ such that B = A_n1F−1A_n2 = ∅, μ(B)≤ μ({x : x_−−i= y_−−i, (x_−, . . . , x_) = (y_−, . . . , y_)}) for some y_−−i, y_−, . . . , y_∈ S. It comes immediately that μ(B)≤ μ(A_n1)≤ λ for some , λ ∈ (0, 1) that are inde-pendent of n₁, n₂. This elucidates that

μ(E)≤ n−1λ→ 0, as n → ∞

for each E ∈n−1_m=0F−mξ.

Furthermore, if  is chosen such that  ≥ [(i − 1)/2], where [x] is the greatest in-teger that is greater than or equal to x. Then ξ_F−1ξ_ = ξ(− − i, ). Inductively, _n−1

m=0F−mξ= ξ(− − (n − 1)i, ). This asserts the lemma.

The following theorem is demonstrated in [27]. An alternative proof is given here for the self-containing of this elucidation.

Theorem 3.2. h_μ(F ) =−ir−1_m=0p_mlog p_m.

Proof. Consider {ξ}∞=1 a sequence of finite partitions of Ω, it is easy to see that

ξ₁⊂ ξ◦ ₂⊂ · · · and◦ ∞_=1ξ_ B, where A⊂ B (respectively A  B) means the σ-algebra◦

generated by A is a subset of (respectively coincides with) that generated by B up to a measure zero set. For each ∈ N, observe that

H_μ(ξ_) =− 

A∈ξ

μ(A) log μ(A)

=−  s−,...,s p_s−· · · p_slog p_s−· · · p_s =−  s−,...,s−1 p_s· · · p_s−1 s p_slog p_s−· · · p_s =− ⎛ ⎝  s−,...,s−1 p_s−· · · p_s−1log p_s−· · · p_s−1+ r−1  m=0 p_mlog p_m ⎞ ⎠ =−(2 + 1) r−1  m=0 p_mlog p_m.

Applying Lemma 3.1 and mathematical induction,

H_μ( n−1_ m=0 F−mξ_) = H_μ(ξ(− − (n − 1)i, )) = −(2 + (n − 1)i + 1) r−1  m=0 p_mlog p_m whenever  is large enough. Hence

h_μ(F ) = lim →∞hμ(F, ξ) =−i r−1  m=0 p_mlog p_m.

This completes the proof.

In general, let f be permutive and depends on x_i, . . . , x_j, where i≤ j, i, j ∈ Z. The measure-theoretic entropy of F can be explicitly expressed.

Corollary 3.3. Denote by i = − min{i, 0} and j = max{j, 0}, then h_μ(F ) = −(i + j) r−1

m=0pmlog pm. More precisely,

(i) if f is leftmost permutive and i < j≤ 0, then h_μ(F ) =−ir−1_m=0p_mlog p_m;

(ii) if f is rightmost permutive and 0≤ i < j, then h_μ(F ) =−jr−1_m=0p_mlog p_m;

(iii) if f is bipermutive, then h_μ(F ) =−(i+ j)r−1_m=0p_mlog p_m.

Proof. It suffices to show that (iii) is true since (i) is derived from Theorem 3.2 and (ii) can be done analogously.

If f is bipermutive, a slight change of the proof of Lemma 3.1 shows thatn−1_m=0F−mξ_=

ξ(− − (n − 1)i,  + (n − 1)j), i.e., ξ_is a generator, for  large enough. Komolgorov-Sinai Theorem indicates that h_μ(F ) = h_μ(F, ξ_), and

H_μ( n−1_ m=0 F−mξ_) =−(2 + (n − 1)(i + j) + 1) r−1  m=0 p_mlog p_m.

This asserts (iii).

Example 3.4. Let S ={0, 1, 2, 3} and let f : S5→ S be defined by

f (x₀, x₁, x₂, x₃, x₄) = 2x₀+ x₃x₀+ x2₁+ 3x₄ mod 4, then f is permutive and i = 0, j = 4. Corollary 3.3 shows that

h_μ(F ) =−4(p₀log p₀+ p₁log p₁+ p₂log p₂+ p₃log p₃) = 4 log 4 whenever uniform Bernoulli measure is considered.

3.2. Topological entropy and topological pressure

In this subsection, the topological entropy of permutive CA is studied. Let X be a metric space and let T : X → X be a continuous transformation. An open cover of X, denote by O, is called a strong generator provided, for any δ > 0, there exists N ∈ N such that n−1_m=0T−mO < δ for all n ≥ N, where A is the diameter of A. In other

words,O is a strong generator if and only if n−1_m=0T−mO → 0, as n → ∞.

Fagnani and Margara [34] demonstrate that bipermutive CA (which is called LRCA in [34]) is a proper subset of expansive CA. D’amico et al. [23] give an algorithm for the computation of expansive CA. We elucidate permutive CA and an explicit formula for the computation of topological entropy is discovered. Notably there exists permutive CA which is not expansive, here we give an example.

Example 3.5. Let S ={0, 1, 2, 3} and let f : S3→ S be defined by f(x₋₂, x₋₁, x₀) =

x₋₂+ 2x2₀ mod 4. Then f is permutive rather than expansive since f2 = x₋₄ which is the local rule of F2 and h_top(F ) = 4 log 2 by Theorem 3.6.

Since a cylinder is open and close, ξ(a, b) is a finite open cover of Ω for a≤ b. The formula of topological entropy of leftmost permutive CA can be expressed in a closed form.

Theorem 3.6. h_top(F ) = i log r.

Proof. The discussion in the proof of Lemma 3.1 indicates thatn−1_m=0F−mξ_= ξ(−− (n− 1)i, ) for  large enough. This demonstrates that cardn−1_m=0F−mξ_= r2+1+(n−1)i and h_top(F, ξ_) = lim n→∞ 1 nlog r 2+1+(n−1)i _{= i log r,}

for  large enough.

The proof is complete since h_top(F ) = lim_→∞h_top(F, ξ_). The following lemma can be done similarly as Lemma 3.1.

Lemma 3.7. Let f be bipermutive and let ξ_ ≡ ξ(−, ) be a finite open cover of Ω for some  ∈ N. Then n−1_m=0F−mξ_ → 0 as n → ∞. In other words, ξ_ is a strong generator.

A strong generator can be used for the calculation of topological entropy.

Theorem 3.8 ([35]). If ξ is a strong generator of an endomorphism (Ω, F ), then h_top(F ) =

h_top(F, ξ).

In general, if f is permutive and depends only on x_i, . . . , x_j, where i ≤ j, i, j ∈ Z, then we can derive h_top(F ) a explicit formula.

Corollary 3.9. Denote by i = − min{i, 0} and j = max{j, 0}, then h_top(F ) = (i + j) log r. More precisely,

(i) if f is leftmost permutive and i < j≤ 0, then h_top(F ) = ilog r; (ii) if f is rightmost permutive and 0≤ i < j, then h_top(F ) = j log r;

(iii) if f is bipermutive, then h_top(F ) = (i + j) log r.

Proof. (i) and (ii) come immediately from Theorem 3.6 while (iii) can be seen similarly as the discussion in the proof of Corollary 3.3.

Remark 3.10. If X is a compact metric space and T : X → X is a continuous trans-formation, variational principle indicates that

h_top(T ) = sup{h_ν(T ) : ν is an ergodic measure}. (8) A measure ν is called a measure with maximum entropy provided h_top(T ) = h_ν(T ). Corollaries 3.3 and 3.9 demonstrate that the uniform Bernoulli measure is a measure with maximum entropy for permutive CA.

For a given potential function φ∈ C(Ω, R), the topological entropy can be generalized to the consideration of topological pressure P (F, φ). Let a₀, a₁, . . . , a_r−1 ∈ R be given

and let φ : Ω→ R be defined by φ(x) = a_x0, we have the following theorem. 8

Theorem 3.11. P (F, φ) = (i− 1) log r + log(ea0_{+ e}a1₊· · · + ear−1_).

Proof. The proof of Lemma 3.1 demonstrates thatn−1_m=0F−mξ_ = ξ(−(n − 1)i − , ) for n∈ N and  large enough. Observe that

p₂(F, φ, ξ_) =  m1,m2  Am1;m2=∅ sup x∈Am1;m2 exp((S₂φ)(x)) =  m1,m2  Am1;m2=∅ exp(φ(x) + φ(F (x))) =  i1,i2∈S

r2+i−1exp(a_i1+ a_i2) = r2+i−1(ea0_{+ e}a1₊· · · + ear−1₎2_,

where A_m1;m2 ∈ [m₁]F−1[m₂] and m₁, m₂ ∈ S2+1. Furthermore, it can be verified that p_n(F, φ, ξ_) = r2+(n−1)(i−1)(ea0_{+ e}a1₊· · ·+ear−1₎n_{, and P (F, φ, ξ) = (i}−1) log r+

log(ea0_{+ e}a1₊· · · + ear−1_).

The proof is completed by taking  to the infinity.

In general, topological pressure of permutive CA can be expressed in a compact form. The proof is similar as above, thus is skipped.

Corollary 3.12. Let the local rule f be permutive and depend on x_m for i ≤ m ≤ j, i, j ∈ Z. Denote by i = − min{i, 0} and j = max{j, 0}. If a₀, a₁, . . . , a_r−1 ∈ R are given, consider a potential function φ ∈ C(Ω, R) defining by φ(x) = a_x0. Then P (F, φ) = (i + j− 1) log r + log(ea0_{+ e}a1₊· · · + ear−1_{). More precisely,}

(i) if f is leftmost permutive and i < j ≤ 0, then h_top(F ) = (i− 1) log r + log(ea0₊ ea1₊· · · + ear−1_);

(ii) if f is rightmost permutive and 0≤ i < j, then h_top(F ) = (j− 1) log r + log(ea0₊ ea1₊· · · + ear−1_);

(iii) if f is bipermutive, then h_top(F ) = (i + j− 1) log r + log(ea0_{+ e}a1₊· · · + ear−1_).

Remark 3.13. If X is a compact metric space and T : X → X is a continuous trans-formation, variational principle for topological pressure says that, for ψ ∈ C(X, R),

P (T, ψ) = sup{h_ν(T ) +

X

ψ dν : ν is an ergodic measure}. (9) A measure ν is called an equilibrium measure provided P (T, ψ) = h_ν(T ) +_Xψ dν.

Corollaries 3.3 and 3.12 help for the determination of an equilibrium measure for permu-tive CA.

Example 3.14. Let f (x₋₁, x₀, x₁) = x₋₁, then P (σ, φ) = log(ea0 _{+ e}a1 ₊· · · + ear−1_),

where a₀, a₁, . . . , a_r−1are given and φ(x) = a_x0 for x∈ Ω. It is easily seen that

h_μ(F ) + Ωφ dμ =− r−1  m=0 p_mlog p_m+ r−1  m=0 a_m· p_m= r−1  m=0 p_m(a_m− log p_m). 9

To determine whether μ is an equilibrium measure, define Φ : [0,∞) → R by Φ(x) =



0, x = 0;

x log x, otherwise. Then Φ is convex and Φ∈ C1((0,∞), R). Moreover,

Φ( n  m=1 α_mx_m)≤ n  m=1 α_mΦ(x_m), for n  m=1 α_m= 1, α_m≥ 0, x_m∈ R. Let α_m= eam_{/λ and x}

m= (pmλ)/eam for 0≤ m ≤ r − 1, where λ =r−1m=0eam.

0 = Φ(1) = Φ( r−1  m=0 α_mx_m) ≤ r−1  m=0 eam λ · p_mλ eam log p_mλ eam = r−1  m=0 p_mlogpmλ eam = log(ea0_{+ e}a1_{+· · · + e}ar−1₎₋ r−1  m=0 p_m(a_m− log p_m).

The equality holds if and only if (p_mλ)/eam _{= 1 for 0}≤ m ≤ r−1, i.e., μ is an equilibrium

measure if and only if p_m= eam_/r−1

m=0eam for 0≤ m ≤ r − 1.

Example 3.15. Consider Wolfram’s rule 150, i.e., S ={0, 1} and f(x₋₁, x₀, x₁) = x₋₁+

x₀+ x₁ mod 2. Let φ : Ω→ R be defined by φ(x) = log p_x0, then P (F, φ) = log 2 and

h_μ(F ) + Ωφ dμ =−2 1  i=0 p_ilog p_i+

Ωlog px0 dμ =−(p0log p0+ p1log p1).

It is easily verified that μ is an equilibrium measure provided p₀= p₁= 1/2.

Furthermore, consider alphabet S ={0, 1, . . . , r − 1} and f and φ are same as above. We conclude that μ is an equilibrium measure provided p_i = 1/r for all i, i.e., uniform Bernoulli measure is an equilibrium measure.

4. Topological properties of permutive cellular automata

This section studies permutive cellular automata in the viewpoint of topological as-pects. A dynamical system is said to be chaotic in the sense of Li-Yorke provided the existence of periodic points with period larger than some given integer and there exists an uncountable set such that any two distinct orbits of it would be arbitrary close but never merge together. The existence of snap-back repeller implies positive topological entropy which is a sufficient condition for the exhibition of Li-Yorke chaos for a given dynamical system.

Definition 4.1. Let X be a metric space and let T : X→ X be continuous, a dynamical system associated with x_n = T (x_n−1) for n ∈ Z+ is said to be chaotic in the sense of Li-Yorke if and only if

a) there exists a positive integer N such that for each integer p ≥ N, T has a point of

period p;

b) there exists a scramble set S, i.e., an uncountable set containing no periodic points

such that

(i) T (S) ⊂ S;

(ii) for every x, y∈ S with x = y, lim sup

m→∞ |T

m_{(x)− T}m_{(y)| > 0, lim inf}

m→∞ |T

m_{(x)− T}m_{(y)| = 0;}

(iii) for every x∈ S and y a periodic point of T , lim sup

m→∞ |T

m_{(x)− T}m_{(y)| > 0;}

For F a cellular automaton, a point z∈ Ω is called an expanding fixed point of F if (i) z is a fixed point of F ;

(ii) there exists > 0 such that for all x ∈ B (z), x= z, |F (x) − F (z)| > |x − z| and

F−m(x)→ z as m → ∞.

The radius such that each x= z is expanding in B (z) is called expanding radius.

Definition 4.2. A point z∈ Ω is called a snap-back repeller if

(i) z is an expanding fixed point of F for some expanding radius ;

(ii) there exists a point x₀ ∈ B (z), x₀ = z, such that FM(x₀) = z for some positive

integer M .

Theorem 4.3. For F a cellular automaton that possesses a snap-back repeller, then F is chaotic in the sense of Li-Yorke.

Instead of giving a theoretical proof, an example is investigated to assert Theorem 4.3 since the proof is quite similar as the one given in [29, 30].

Example 4.4. Consider the alphabetA = {0, 1}. Wolfram develops a qualitative clas-sification scheme of 256 elementary one-dimensional CA rules that distinguishes four dif-ferent complexity classes [36]. Cattaneo et al. [19] shows that the global maps F₁₀₂ and

F₁₇₀ are topological conjugate at one-sided CA, Σ+₂ ={x = (x_i)_i≥0: x_i ∈ A for all i}, where the local rules of F₁₀₂ and F₁₇₀ are

f₁₀₂:A3→ A defined by f₁₀₂(x₋₁, x₀, x₁) = x₀+ x₁ mod 2, and

f₁₇₀:A3→ A defined by f₁₇₀(x₋₁, x₀, x₁) = x₁,

respectively. Since F₁₇₀possesses rich dynamical behavior [37], F₁₀₂is thus chaotic. Here we show that F₁₀₂ exhibits a snap-back repeller.

A standard metric d on Σ+₂ is defined by

d(x, y) = ∞  i=0 |xi− yi| 2i . (10)

Let z = 0∞ and = 1/2, then z is a fixed point. For each x∈ B (z), x₀= x₁= 0. It is easily seen that

(i) d(F₁₀₂(x), z) > d(x, z) for all x∈ B (z); (ii) F₁₀₂−k(x)→ z as k → ∞ for all x ∈ B (z);

(iii) let y = (0011001100110011 . . .), then y∈ B (z), F₁₀₂(y) /∈ B (z) and F₁₀₂3 (y) = z. That is, z is a snap-back repeller. Theorem 4.3 indicates that F₁₀₂is chaotic in the sense of Li-Yorke.

A significant phenomenon of Li-Yorke chaos is there exists N ∈ N such that, for

n ≥ N, F exhibits an n-periodic orbit. For this example, we can examine more about

the periodic points.

Let ς = 2−4. It is easily verified F₁₀₂|_Bς(z) is well-defined and B_ς(z) = ₀[00000]₄. For simplicity, the left index of a cylinder is omitted if it is zero, i.e., denote ₀[· · · ]_mby [· · · ]_m.

Set G = F−3|_Bς(z)and Q = G(B_ς(z)), then Q = [00110011]₇, F (Q) = [0101010]₆and

F2(Q) = [111111]₅ are compact subsets of the complement of B (z) = [00]₁. Observe that

F−1Q = [000100010]₈, F−2Q = [0000111100]₉, F−3Q = [00000101000]₁₀.

This implements that F−3(x)∈ B_ς(z) for all x∈ Q. More than that, F−kQ⊂ B_ς(z) for all k≥ 3. Notably, Q = G(B_ς(z)) = F−3(B_ς(z)) and

F−k◦ G : B_ς(z)→ B_ς(z), for k≥ 3. (11) Brouwer’s fixed point theorem asserts that there exists y_k ∈ B_ς(z) such that (F−k ◦

G)(y_k) = y_k, i.e., Fk+3(y_k) = y_k, for all k≥ 3.

It remains to show that y_k is actually of period k + 3.

Since Fk(y_k) = G(y_k)∈ Q, F is expanding in B_ς(z) indicates that

Fn(y_k)= y_k, for 1≤ n ≤ k.

Also, F (Q), F2(Q) ⊂ B (z)c implies Fn(y_k)= y_k for n = k + 1, k + 2. Hence, y_k is of period k + 3.

As a conclusion, let N = 6. For all n ≥ N, there exists y_n ∈ B_ς(z) such that

Fn(y_n) = y_n and Fk(y_n)= y_n for 1≤ k ≤ n − 1.

Next, we demonstrate an algorithm so that all the periodic points can then be deter-mined.

Consider p(x) = 1+x the polynomial over finite fieldZ₂corresponding to the local rule

f₁₀₂(x₀, x₁) = x₀+ x₁ mod 2. Then p2(x) = 1 + x2is corresponding to f₁₀₂2 (x₀, x₁, x₂) =

x₀+ x₂ mod 2, which is the local rule of F₁₀₂2 . Inductively we have pn(x) = (1 + x)n= _n

i=0aixi is corresponding to f102n (x0, . . . , xn) = ni=0aixi mod 2, which is the local

rule of F₁₀₂n . Using this periodic points can be written exactly.

For instance, consider n = 6. Then p6(x) = 1 + x2 + x4 + x6, this indicates

f₁₀₂6 (x₀, . . . , x₆) = x₀+ x₂+ x₄+ x₆ mod 2. If y∈ B (z) is of period 6, then

y_i= F6(y)_i= y_i+ y_i+2+ y_i+4+ y_i+6, for all i∈ Z+.

That is,

y_i+2+ y_i+4+ y_i+6= 0 mod 2, for all i∈ Z+.

This also implies y_i+2= y_i+8 for all i∈ Z+. Since y∈ B_ς(z) = [00000]₄, we have

y_i= 

1, 6|i or 6|(i − 2); 0, otherwise. is uniquely determined and y is exactly of period 6.

To construct the scramble set, denote by U = F2(Q), V = B_ς(z). Then U and V are both compact. It can be verified without difficulty that

(i) U∩ V = ∅; (ii) F6(U )⊃ V ; (iii) F6(V )⊃ U ∪ V .

LetE be the the collection of sequences E = {E_n}∞_n=1 satisfying (i) either E_n= U or E_n ⊂ V , and F (E_n)⊃ E_n+1;

(ii) if E_n = U , then n = m2 for some m∈ N and E_n+1, E_n+2⊂ V .

Let card(E, n) be the cardinality of i’s such that E_i = U for 1 ≤ i ≤ n. For each

w∈ (0, 1), pick Ew={E_nw}∞_n=1to be a sequence inE such that

lim

n→∞

card(Ew, n2)

n = w.

Then E = {Ew ∈ E : w ∈ (0, 1)} is uncountable, and F6(E_nw) ⊃ E_n+1w . Therefore, for each Ew ∈ E, there exists x_w ∈ U ∪ V such that F6n(x_w) ∈ E_nw for n ∈ N. Let

K = {x_w : w ∈ (0, 1)} be the collection of these points, then S is also uncountable. Moreover, K satisfies the following conditions:

(i) K contains no periodic points of F ; (ii) F (K)⊂ F6(K)⊂ K;

(iii) For x= y ∈ K, there exist infinitely many n’s such that F6n(x)∈ U, F6n(y)∈ V . (For the details, we refer reader to [28].) U and V are both compact implies inf{|x − y| :

x∈ U, y ∈ V } > 0. That is,

lim sup

m→∞ |F

k_{(x)− F}k_{(y)| > 0, for all x = y ∈ K. (12)}

For y a periodic point of F . y /∈ U ∪ V . Since the union of compact sets is still compact,

lim sup

k→∞ |F

k_{(x)− F}k_{(y)| > 0, for all x ∈ K. (13)}

Denote D_n = F−6n(B_ς(z)) for n≥ 0. For each δ > 0, there exists J = J(δ) > 0 such that

|x − z| < δ

2, for all n≥ J, x ∈ Dn. (14)

Let E ⊂ E be the collection of Ew={E_nw}∞_n=1such that if E_nw= U for both n = m2and

n = (m + 1)2, then Ew_n = D_2m−k for n = m2+ k, where k = 1, 2, . . . , 2m. Otherwise,

E_nw⊂ V .

Similarly, F6(E_nw) ⊃ E_n+1w and there exists x_w ∈ U ∪ V such that F6n(x_w) ∈ Ew_n for n∈ N. Let S be the collection of x_w, then S ⊂ K is uncountable. Also, for every

s = t ∈ (0, 1), there exists infinitely many m’s such that F6n(x_s) ∈ E_ns = D_2m−1 and

F6n(x_t)∈ E_nt = D_2m−1, where n = m2+ 1. (14) elucidates that|x − z| < δ/2 for all

x∈ D_2m−1when m large enough. Hence, for any δ > 0,

|Fk_(x

t)− Fk(xs)| ≤ |Fk(xt)− z| + |Fk(xs)− z| < δ, for k large enough.

In other words,

lim inf

k→∞ |F

k_{(x)− F}k_{(y)| = 0, for all x, y ∈ S. (15)}

The construction of scramble setS is done. This completes the example.

Remark 4.5. It is well-known that the topological entropy of F₁₀₂ is log 2. Blanchard et al. [32] shows that the positive topological entropy implies chaos in the sense of Li-Yorke. This asserts Theorem 4.3 in the quantitative viewpoint.

Proposition 4.6. Each bipermutive CA exhibits a snap-back repeller.

Proof. Fagnani and Margara [34] indicates that a bipermutive CA is topological con-jugate to a one-sided shift. It is easy to verify that it exhibits a snap-back repeller.

The proof is complete.

Notably that bipermutivity is optimized for the exhibition of snap-back repellers when two-sided CA is considered. The following is a counterexample.

Proposition 4.7. Consider F₁₀₂ a two-sided CA, then F₁₀₂ exhibits no snap-back re-pellers.

Proof. Let p(x) = 1 + x be the polynomial corresponding to f102. It can be easily

verified p2n(x) = 1 + x2n for all n∈ N, i.e., f₁₀₂2n(x₀, . . . , x₂n) = x₀+ x₂n mod 2.

If y ∈ AZ satisfies F₁₀₂2n(y) = y. Then y = 0∞, which is a fixed point. This means

F₁₀₂ has no period 2n points for all n ∈ N. Hence F₁₀₂ can never exhibit a snap-back repeller.

5. Conclusion

In this dissertation, we elucidate the following results.

a) The formula of topological entropy of permutive cellular automata can be written down in a compact form. Ward [24] studies the topological entropy of additive cellular automata whose cardinality of alphabet is prime, Corollary 3.9 generalizes his work. b) D’amico et al. [23] investigate an algorithm for the computation of topological entropy

for expansive cellular automata. Our result works for those cellular automata which are permutive but non-expansive.

c) The measure-theoretic entropy and topological pressure of permutive cellular au-tomata is demonstrated and the uniform Bernoulli measure is a measure with max-imum entropy. More than that, Parry measure is an equilibrium measure for some potential function.

d) Bipermutive cellular automata exhibit snap-back repeller, which is a sufficient condi-tion for the determinacondi-tion of Li-Yorke chaos. Furthermore, this condicondi-tion is optimized for two-sided shift space.

Acknowledge

The authors thank Prof. Song-Sun Lin for his valuable suggestion during the prepa-ration of this work. The first author is partially supported by NSC grant 98-2628-M-259-001, the second author is partially supported by the National Center for Theoretical Sciences in Taiwan, and the third author thanks NSC grant 97-2815-C-026-001-M for the partially support.

References

[1] S. Ulam, Random process and transformations, Proc. Int. Congress of Math. 2 (1952) 264–275. [2] J. von Neumann, Theory of self-reproducing automata, Univ. of Illinois Press, Urbana, 1966. [3] J. M. Greenberg, S. P. Hastings, Spatial patterns for discrete models of diffusion in excitable media,

SIAM J. Appl. Math. 34 (1978) 515–523.

[4] G. Bub, A. Shrier, L. Glass, Spiral Wave Generation in Heterogeneous Excitable Media, Phys. Rev. Lett. 88 (2002) 058101.

[5] Y. B. Chernyak, A. B. Feldman, R. J. Cohen, Correspondence between discrete and continuous models of excitable media: Trigger waves, Phys. Rev. E 55 (1997) 3125–3233.

[6] A. B. Feldman, Y. B. Chernyak, R. J. Cohen, Wave-front propagation in a discrete model of excitable media, Phys. Rev. E 57 (1998) 7025–7040.

[7] J. Hardy, O. de Pazzis, Y. Pomeau, Molecular dynamics of a classical lattice gas: transport prop-erties and time correlation functions, Phys. Rev. A 13 (1976) 1949–1960.

[8] M. Gardner, Wheels, life and other mathematical amusements, W. H. Freeman and Company, 1983. [9] J. M. Greenberg, B. D. Hassard, S. P. Hastings, Pattern Formation and Periodic Structure in Systems Modeled By Reaction-Diffusion Equations, Bull. Amer. Math. Soc. 84 (1978) 1296–1327. [10] J. M. Greenberg, C. Greene, S. P. Hastings, Acombinatorial problem arising in the study of

reaction-diffusion equations, SIAM J. Algebraic Discrete Methods 1 (1980) 34–42.

[11] D. Richardson, Tessellation with Local Transformations, J. Compuut. System Sci. 6 (1972) 373V388. [12] A. R. Smith III, Simple Computational Universal Spaces, J. Assoc. Comput. Mach. 18 (1971)

339V353.

[13] G. Y. Vichniac, Boolean Derivatives on Cellular Automata, Phys. D 45 (1990) 63V74.

[14] J. R. Weimar, J. J. Tyson, L. T. Watson, Third generation cellular automaton for modeling excitable media, Phys. D 55 (1992) 328–339.

[15] S. Wolfram, Statistical mechanics of cellular automata, Rev. Modern Physics 55 (1983) 601–644. [16] G. A. Hedlund, Endomorphisms and automorphisms of full shift dynamical system, Math. Systems

Theory 3 (1969) 320–375.

[17] S. Wolfram, Computation theory of cellular automata, Comm. Math. Phys. 96 (1984) 15–57. [18] S. Wolfram, Twenty problems in the theory of cellular automata, Phys. Scripta 9 (1985) 170–172. [19] G. Cattaneo, M. Finelli, L. Margara, Investigating topological chaos by elementary cellular

au-tomata dynamics, Theoret. Comput. Sci. 244 (2000) 219–241.

[20] B. Codenotti, L. Margara, Transitive Cellular Automata are Sensitive, Amer. Math. Monthly 103 (1996) 58V62.

[21] P. Favati, G. Lotti, L. Margara, Additive one-dimensional cellular automata are chaotic according to Devaney’s definition of chaos, Theor. Comput. Sci. 174 (1997) 157–170.

[22] L. P. Hurd, J. Kari, K. Culik, The topological entropy of cellular automata is uncomputable, Ergodic Theory Dynam. Systems 82 (1992) 255–265.

[23] M. D’amico, G. Manzini, L. Margara, On computing the entropy of cellular automata, Theor. Comput. Sci. 290 (2003) 1629V1646.

[24] T. Ward, Additive cellular automata and volume growth, Entropy 2 (2000) 142–167.

[25] M. A. Shereshevsky, Ergodic properties of certain surjective cellular automata, Monatsh. Math. 114 (1992) 305–316.

[26] D. A. Lind, Application of ergodic theory and sofic systems to cellular automata, Physica D 10 (1984) 36–44.

[27] M. Sablik, Measure rigidity for algebraic bipermutive cellular automata, Ergodic Theory Dynam. Systems 27 (2007) 1965–1990.

[28] T.-Y. Li, J. A. Yorke, Period three implies chaos, Am. Math. Monthly 82 (1975) 985–992. [29] F. R. Marotto, Snap-back repellers imply chaos in_Rn, J. Math. Anal. Appl. 63 (1978) 199–223.

[30] F. R. Marotto, On redefining a snap-back repeller, Chaos Solitons Fractals 25 (2005) 25–28. [31] C. B. Garc´ıa, Chaos and topological entropy in dimensionn > 1, Ergodic Theory Dynam. Systems

6 (1986) 163–165.

[32] F. Blanchard, E. Glasner, S. Kolyada, A. Maass, On Li-Yorke pairs, J. Reine Angew. Math. 547 (2002) 51–68.

[33] P. Walters, An introduction to ergodic theory, Springer-Verlag New York, 1982.

[34] F. Fagnani, L. Margara, Expansivity, permutivity, and chaos for cellular automata, Theory Comput. Systems 31 (1998) 663–677.

[35] M. Pollicott, M. Yuri, Dynamical systems and ergodic theory, Cambridge University Press, 1998. [36] S. Wolfram, A new kind of science, Wolfram Media, Champaign Illinois, USA, 2002.

[37] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer-Verlag, Berlin, NY, 1990.