The Complexity of Permutive Cellular Automata

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The complexity of permutive cellular automata

JUNG-CHAOBAN1,2, CHIH-HUNGCHANG3?, TING-JUCHEN3, MEI-SHAOLIN3

1

Department of Applied Mathematics, National Dong Hwa University, Hualian 97063, Taiwan, R.O.C.

2

Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan, R.O.C.

3

Department of Mathematics, National Central University, Chung-Li 32054, Taiwan, R.O.C.

This paper studies one-dimensional permutive cellular automata in two aspects: Ergodic and topological behavior. Through in-vestigating measure-theoretic entropy and topological pressure, we show taht Parry measure is the unique equilibrium measure whenever the potential function depends on one coordinate. In other words, permutive cellular automata exhibit no phase tran-sition. Furthermore, the existence of snap-back repellers for a cellular automaton infers Li-Yorke chaos and bipermutive cellu-lar automata guarantee the subsistence of snap-back repellers.

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

1 INTRODUCTION

Cellular automaton (CA), introduced by Ulam [18] and Neumann [19] 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 [3, 4, 6, 9, 8, 10, 15, 17, 21].

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The present paper devotes to study the complexity of permutive CA (de-fined later) in two aspects, say, the viewpoints of thermodynamics and topo-logical behaviors.

Let (X, µ) be a measurable space and let T : X → X be a continuous transformation. Given a potential function φ : X → R, the pressure function P (T, φ) of T with respect to φ indicates the energy of the system. Varia-tional principle stands that P (T, φ) = sup{h(µ) +R φdµ}, where h(µ) is the measure-theoretic entropy and the supremum is taken for ergodic mea-sures. A measure that attains the supremum is called an equilibrium measure. The uniqueness of equilibrium measure asserts that there exists no phase tran-sition in this system while the existence of two or more equilibrium measures implies phase transition may occur.

Investigating the formula of topological pressure and measure-theoretic entropy help for the determination of number of equilibrium measures. In other words, we want to answer whether permutive CA exhibit phase transi-tion or not.

We demonstrate the formulae of measure-theoretic entropy (Theorem 3.1) and topological pressure (Theorem 3.4). The measure-theoretic entropy of permutive CA is also elucidated in [16]. This thesis gives it an alternative proof. Moreover, consider µ = (p0, . . . , pr−1) a Bernoulli measure and the potential function φ is given by φ(x) = log px0 for x ∈ Ω, where r is the

number of alphabet S and Ω = SZ_{is the space of bi-infinite sequence. The} permutivity of CA asserts that Parry measure is the unique equilibrium mea-sure, thus there is no phase transition in such system (Corollary 3.6).

The second part studies the complexity of the topological behavior ex-hibited by permutive CA. Li and Yorke [12] discover that, if a first-order difference equation

xi+1 = f (xi), i ∈ Z+,

where xi ∈ 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 demonstrates the existence of a snap-back repeller implements Li-Yorke chaos [13, 14]. Garc´ıa also shows the existence of snap-back repeller admits positive topological entropy, which is a sufficient condition for Li-Yorke chaos [7, 2].

This is a motivation that, in CA, does the existence of snap-back repeller also implements Li-Yorke chaos? The answer is affirmative (Theorem 4.1).

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Notably, the dynamical system considered in [13] is differentiable. CA, how-ever, is only a continuous system. Furthermore, each bipermutive CA exhibits a snap-back repeller (Proposition 4.3) and there is an example that permutive but not bipermutive CA is not Li-Yorke chaotic.

The rest of this elucidation is organized as follows. Section 2 gives some notations and definitions. Section 3 studies the ergodic properties of per-mutive 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.

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 = (xn)∞−∞. Hedlund examines CA in the viewpoint of symbolic dynamical systems [11]. He shows that 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 (xi−k, . . . , xi+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 local rule f : S2k+1 → S is called 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 xiwhenever the other variables are fixed; (ii) f does not depend on xjfor j < i (respectively j > i).

f is called bipermutive provided f is both leftmost and rightmost permutive. The family of permutive cellular automata consists of the following three types of local rules.

1. f is leftmost permutive and does not depend on xifor i > 0; 2. f is rightmost permutive and does not depend on xifor i < 0; 3. f is bipermutive.

For reader’s convenience, we recall definitions of measure-theoretic en-tropy, topological enen-tropy, and topological pressures. Reader may refer to [20] for more details.

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Let µ be an invariant probability measure on (Ω, F ) and let α and β be two finite measurable partitions of Ω. Define αW β and Hµ(α) by

α_β = {A\B : A ∈ α, B ∈ β} and

Hµ(α) = − X

A∈α

µ(A) log µ(A), respectively. The measure-theoretic entropy of F is defined by

hµ(F ) = sup ( lim n→∞ 1 nHµ( n−1 _ m=0 F−mα) ) , (1)

where the supremum is taken over all finite measurable partitions α. Define d : Ω × Ω → R by d(x, y) = ∞ X i=−∞ |xi− yi| r|i| , x, y ∈ Ω. (2)

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. Thena[sa, . . . , sb]b is not only open but close in Ω.

Let P be an open cover of Ω, denote by H(P) = inf{log # ˆP},

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

htop(F ) = sup ( lim n→∞ 1 nH( n−1 _ m=0 F−mP) ) , (3)

where the supremum is taken over all open covers P.

In addition, for α an open cover of Ω and φ ∈ C(Ω, R) a continuous function from Ω to R, denote by

pn(F, φ, α) = inf    X B∈β sup x∈B e(Snφ)(x)_{: β is a finite subcover of} n−1 _ m=0 F−mα    , where n ∈ N and Snφ =P n−1 m=0φ ◦ F m_{. Define} P (F, φ) = lim sup δ→0 lim n→∞ 1 nlog pn(F, φ, α) : diam(α) ≤ δ . (4)

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

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

1. there exists a positive integer N such that for each integer p ≥ N , T has a point of period p;

2. there exists a scramble set S, i.e., an uncountable set containing no periodic points such that

(a) T (S) ⊂ S;

(b) for every x, y ∈ S with x 6= y, lim sup

m→∞

|Tm(x) − Tm(y)| > 0, lim inf m→∞ |T

m_{(x) − T}m_{(y)| = 0;}

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

m→∞

|Tm_{(x) − T}m_{(y)| > 0;}

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

1. z is a fixed point of F ;

2. there exists > 0 such that for all x ∈ B(z), x 6= z, |F (x) − F (z)| > |x − z| and F−m_{(x) → z as m → ∞.}

The radius such that each x 6= z is expanding in B(z) is called expanding radius.

Definition 2.1. A point z ∈ Ω is called a snap-back repeller if 1. z is an expanding fixed point of F for some expanding radius ; 2. there exists a pointx0 ∈ B(z), x0 6= z, such that FM(x0) = z for

some positive integerM .

3 EQUILIBRIUM MEASURES OF PERMUTIVE CELLULAR

AU-TOMATA

This section investigates equilibrium measures of permutive CA through study-ing the measure-theoretic entropy and topological pressure. Ban and Chang

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[1] show that, if the potential function depends on one coordinate, uniform Bernouli measure is an equilibrium measure for linear CA with prime-state. We extends their result to permutive CA.

In the rest of this investigation, local rule f is permutive and depends on xi, . . . , xj, where i ≤ j and i, j ∈ Z.

3.1 Measure-theoretic entropy

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

µ(a[sa, . . . , sb]b) = psa· · · psb, fora[sa, . . . , sb]b⊂ Ω.

Denote by bi = − min{i, 0} and bj = max{j, 0}. The following theorem is also demonstrated in [16]. Here we give an alternative proof.

Theorem 3.1. If f is permutive, then

hµ(F ) = −(bi + bj) r−1 X m=0

pmlog pm.

Before giving proof, we introduce a lemma. For ` ∈ N, denote by ξ` = {−`[s−`, . . . , s`]`: s−`, . . . , s`∈ S} a measurable partition of Ω.

Lemma 3.2. If f is permutive, then n−1

_ m=0

F−mξ`= ξ(−` − (n − 1)ˆi, ` + (n − 1)ˆj)

provided` large enough, where ξ(a, b) = {a[xa, . . . , xb]b: xa, . . . , xb∈ S}. Proof. We discuss the case that f is permutive of type 1, i.e., ˆi = −i > 0 and ˆj = 0, the other cases can be done via analogous argument. First observe that for each z = (za, . . . , zb) ∈ Sb−a+1, f_b−a+i−1 z ∈ Sb−a−i+1, where fm: S−i+m+1 → Sm+1is defined by

fm(xi−m, . . . , x0) = (f (xi−m, . . . , x−m), . . . , f (xi, . . . , x0)), for m ∈ N, and f0 = f . Denote fb−a+iby f without ambiguity. For sb+i+1, . . . , sb ∈ S, the leftmost permutivity of f at xi implies there admits a unique ˜zb+i such that f (˜zb+i, sb+i+1, . . . , sb+j) = zb. Repeating this process there are uniquely determined ˜zb+i−1, . . . , ˜za+i∈ S such that

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Denote by ξ`= {Al}r

2`+1

l=1 , above discussion and induction asserts that F−mAn1

\

F−mAn2 = ∅, for n16= n2, m ∈ Z

+_.

Furthermore, if ` is chosen such that ` ≥ [i/2], where [x] is the greatest integer that is less than or equal to x. Then ξ`W F−1ξ` = ξ(−` + i, `). Inductively,Wn−1

m=0F−mξ`= ξ(−`+(n−1)i, `). This asserts the lemma. Proof of Theorem 3.1. The case that f is permutive of type 1 is proved, the other cases can be done similarly. Consider {ξ`}∞`=1 a sequence of finite partitions of Ω, it is easy to see that ξ1

◦ ⊂ ξ2

◦

⊂ · · · andW∞

`=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

Hµ(ξ`) = − X

A∈ξ`

µ(A) log µ(A)

= − X s−`,...,s`−1 ps−`· · · ps`−1 X s` ps`log ps−`· · · ps` = −(2` + 1) r−1 X m=0 pmlog pm.

Applying Lemma 3.2 and mathematical induction,

Hµ( n−1 _ m=0 F−mξ`) = −(2` − (n − 1)i + 1) r−1 X m=0 pmlog pm

whenever ` is large enough. Hence

hµ(F ) = lim `→∞hµ(F, ξ`) = i r−1 X m=0 pmlog pm.

This completes the proof.

Example 3.3. 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, thenf is permutive of type 2 and bi = 0, bj = 4. Theorem 3.1 shows that

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3.2 Topological pressure

Let φk : Sk → R be given and let φ : Ω → R be defined by φ(x) = φk(x0· · · xk−1). Set ξ ≡ ξ(0, k − 1) a measurable partition of Ω. Define transition matrix Tφ,F = (tmn)1≤m,n≤rkwith respect to φ by

tmn= dmnexp φ(x), x ∈ Cm, (5) where Cm= [c0, . . . , ck−1] ∈ ξ, m = 1 +P k−1 `=0c`· rk−`−1and dmn= 1, Cm∩ F−1Cn 6= ∅; 0, otherwise.

We have the following theorem.

Theorem 3.4. If f is permutive, then the topological pressure P (F, φ) = s log r + log ρ, where ρ is the spectral radius of Tφ,F ands = max{0, ˆi + ˆ

j − k}.

Proof. The case f is permutive of type 1 is considered while the other cases can be proved similarly.

Without loss of generality we may assume k = 2. Let apq ∈ R, 0 ≤ p, q ≤ r − 1, be given and let φ : Ω → R be defined by φ(x) = ax0x1.

Set ξ = {0[pq]1 : 0 ≤ p, q ≤ r − 1} ≡ {C1, . . . , Cr2}. Define eT_φ,F =

(tmn)1≤m,n≤r2, where tmn= (smnexp apq), m = pr + q + 1 and

smn=

rs, Cm∩ F−1Cn6= ∅; 0, otherwise.

Let Dp = #{q : Cp∩ F−1Cq 6= ∅}. Then Dp= Dq for 1 ≤ p, q ≤ r2and thus can be denoted by a constant D. Observe that

p1(F, φ, ξ) = X i1,i2∈S

exp(ai1i2) = | eTφ,F|/(D · r

s_),

where |A| =P amnis the 1-norm for nonnegative matrix A. Moreover, p2(F, φ, ξ) = X m1,m2∈S2 X A_m1;m26=∅ exp(φ(x) + φ(F (x))) = | eT2_φ,F|/(D · rs_),

where the summation is taken for all connected Am1;m2 ∈ [m1]∩F

−1_[m 2]. It comes from induction that pn(F, φ, ξ) = | eTnφ,F|/(D · r

s

) for n ∈ N, Perron-Frobenius theorem demonstrates that P (F, φ, ξ) = log ˜ρ = s log r + log ρ, where ˜ρ is the spectral radius of eTφ,F and ρ is the spectral radius of Tφ,F.

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Fix N ∈ N and set ξN = ξ(−N, N ). It can be verified that p1(F, φ, ξN) = r2(N −1)(D · rs)−1| eTφ,F| and pn(F, φ, ξN) = r2(N −1)(D · rs)−1| eTnφ,F| for n ∈ N. This infers that P (F, φ, ξN) = log ˜ρ = s log r + log ρ for all N ∈ N. The proof is done by letting N tend to infinity.

If potential function φ depends on only one coordinate, that is, without loss of generality we may assume φ : Ω → R is defined by φ(x) = ax0,

where a0, a1, . . . , ar−1 ∈ R are given. Theorem 3.4 can be expressed in an explicit form.

Corollary 3.5. Let a0, a1, . . . , ar−1 ∈ R be given and let φ : Ω → R be defined byφ(x) = ax0. Iff is permutive, then P (F, φ) = (ˆi + ˆj − 1) log r +

log(ea0_{+ e}a1_{+ · · · + e}ar−1_).

Proof. Let ξ = {[0], . . . , [r − 1]} be the standard measurable partition of Ω. Permutivity of f asserts ˆi+ˆj ≥ 1 and thus the kth row of the transition matrix Tφ,F ∈ Mr(R) is eak−1(1 1 · · · 1) for 1 ≤ k ≤ r. Applying Theorem 3.4 derives the desired result.

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{hν(T ) + Z

X

ψ dν : ν is an ergodic measure}. (6) A measure ν is called an equilibrium measure provided P (T, ψ) = hν(T ) + R

Xψ dν. Theorems 3.1 and 3.4 help for the determination of equilibrium measures of permutive CA.

Corollary 3.6. If f is permutive and potential function φ is given as in Corol-lary 3.5, then Parry measure is the unique equilibrium measure. In other word, such cellular automaton possesses no phase transition.

Proof. If f is permutive and ˆi+ ˆj = 1, then P (F, φ) = log(ea0_{+ e}a1_{+ · · · +}

ear−1_{). Moreover,} hµ(F )+ Z Ω φ dµ = − r−1 X m=0 pmlog pm+ r−1 X m=0 am·pm= r−1 X m=0 pm(am−log pm).

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

Φ(x) =

0, x = 0;

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Then Φ is convex and Φ ∈ C1_{((0, ∞), R). Moreover,} Φ( n X m=1 αmxm) ≤ n X m=1 αmΦ(xm), for n X m=1 αm= 1, αm≥ 0, xm∈ R. Let αm = eam/λ and xm = (pmλ)/eam for 0 ≤ m ≤ r − 1, where λ =Pr−1 m=0e am_. 0 = Φ(1) = Φ( r−1 X m=0 αmxm) ≤ r−1 X m=0 eam λ · pmλ eam log pmλ eam = r−1 X m=0 pmlog pmλ eam = log(ea0_{+ e}a1_{+ · · · + e}ar−1_{) −} r−1 X m=0 pm(am− log pm). The equality holds if and only if (pmλ)/eam = 1 for 0 ≤ m ≤ r − 1, i.e., µ is an equilibrium measure if and only if pk = eak/P

r−1 m=0e

am _for

0 ≤ k ≤ r − 1.

When ˆi + ˆj ≥ 2, then P (F, φ) = (ˆi + ˆj − 1) log r + logP eam _and

hµ(F ) = −(ˆi + ˆj)P pmlog pm. The factP pmlog p−1m ≤ log r and the equality holds if and only if pk= p`for k 6= ` implements that

−(ˆi + ˆj)Xpmlog pm+ X

pmam≤ (ˆi + ˆj) log r + log X

eam_.

Moreover, the equality holds if and only if pk=

eak

P eam and pk = p`, for 0 ≤ k, ` ≤ r − 1.

The proof is complete.

4 TOPOLOGICAL PROPERTIES OF PERMUTIVE CELLULAR

AU-TOMATA

This section studies permutive cellular automata in the viewpoint of topologi-cal aspects. A dynamitopologi-cal 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.

We show that, for a cellular automaton, the existence of snap-back repeller implies the exhibition of Li-Yorke chaos and bipermutive cellular automata possesses a snap-back repeller.

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Theorem 4.1. If F is a cellular automaton that possesses a snap-back re-peller, thenF is chaotic in the sense of Li-Yorke.

Instead of giving a theoretical proof, an example is investigated to assert Theorem 4.1 since the proof is similar as the argument given in [13, 14]. Example 4.2. Consider F Wolfram’s rule 102 on Σ+₂ = {x = (xi)i≥0 : xi ∈ {0, 1} for all i}, where the local rules f : {0, 1}3 → {0, 1} is defined byf (x−1, x0, x1) = x0+ x1 mod 2 and the metric d on Σ+2 is defined by

d(x, y) = ∞ X i=0 |xi− yi| 2i . (7)

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

(1) d(F102(x), z) > d(x, z) for all x ∈ B(z); (2) F₁₀₂−k(x) → z as k → ∞ for all x ∈ B(z);

(3) lety = (0011001100110011 . . .), then y ∈ B(z), F102(y) /∈ B(z) andF3

102(y) = z.

That is,z is a snap-back repeller.

To show that_{F is Li-Yorke chaotic, we need to find N ∈ N such that, for} n ≥ N , F exhibits an n-periodic orbit and show the existence of scramble setS.

Letς = 2−4. It is easily verifiedF |Bς(z) is well-defined andBς(z) =

0[00000]4. SetG = F−3|Bς(z) and Q = G(Bς(z)), then Q, F (Q) and

F2_{(Q) are compact subsets of the complement of B}

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

F−k◦ G : Bς(z) → Bς(z), fork ≥ 3. (8)

Brouwer’s fixed point theorem asserts that there existsyk ∈ Bς(z) such that (F−k◦ G)(yk) = yk, i.e.,Fk+3(yk) = yk, for allk ≥ 3.

It remains to show thatykis actually of periodk + 3. SinceFk_(y

k) = G(yk) ∈ Q, F is expanding in Bς(z) indicates that Fn(yk) 6= yk, for1 ≤ n ≤ k.

Also,F (Q), F2_{(Q) ⊂ B}

(z)c impliesFn(yk) 6= yk forn = k + 1, k + 2. Hence,ykis of periodk + 3.

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As a conclusion, letN = 6. For all n ≥ N , there exists yn ∈ Bς(z) such thatFn(yn) = ynandFk(yn) 6= ynfor1 ≤ k ≤ n − 1.

The construction of the scramble set is similar as the method in [12], thus is skipped.

This completes the example.

Proposition 4.3. Each bipermutive cellular automaton exhibits a snap-back repeller.

Proof. Fagnani and Margara indicate that a bipermutive CA is topological conjugate to a one-sided shift [5]. 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 re-pellers when two-sided CA is considered. The following is a counterexample. Example 4.4. Wolfram’s rule 102 exhibits no snap-back repellers on Σ2 = {x = (xi)i∈Z: xi∈ {0, 1} for all i}.

Proof. For n ∈ N and y ∈ Σ2 satisfies F2

n

(y) = y. It is easily seen that y = 0∞_{, which is a fixed point. This means F has no period 2}n_{points for all} n ∈ N. Hence F can never exhibit a snap-back repeller.

5 ACKNOWLEDGE

The authors thank Prof. Song-Sun Lin for his valuable suggestion during the preparation 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.

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