Evaluating the Quality of Local Structure Approximation Using Elementary Rule 14
EEvaluating the quality of local structureapproximation using elementary rule 14
Henryk Fuk´s and Francis Kwaku Combert
Department of Mathematics and Statistics, Brock University,St. Catharines, ON, Canada [email protected] , [email protected] Abstract.
Cellular automata (CA) can be viewed as maps in the spaceof probability measures. Such maps are normally infinitely-dimensional,and in order to facilitate investigations of their properties, especially inthe context of applications, finite-dimensional approximations have beenproposed. The most commonly used one is known as the local structuretheory, developed by H. Gutowitz et al. in 1987. In spite of the popularityof this approximation in CA research, examples of rigorous evaluationsof its accuracy are lacking. In an attempt to fill this gap, we constructa local structure approximation for rule 14, and study its dynamics ina rigorous fashion, without relying on numerical experiments. We thencompare the outcome with known exact results.
Keywords: rule 14, local structure approximation, invariant manifolds
One-dimensional elementary cellular automata (CA) can be viewed as mapsin the space of probability measures over bi-infinite binary sequences (to becalled configurations ). This can be understood as follows. Suppose that we startwith a large set of initial configurations drawn from a certain distribution (forexample, from the Bernoulli distribution). Let us now suppose that we applya given cellular automaton rule to all these configurations. The resulting set ofconfigurations is usually no longer described by Bernoulli distribution, but bysome other distribution. We can thus say that the CA rule transforms the initialprobability measure into some other measure, and when we apply the local ruleagain and again, we obtain a sequence of measures, to be called the orbit of theinitial measure.This approach, however, is not without difficulties. In order to fully describea probability measure over bi-infinite binary sequences, one needs to specifyinfinitely many block probabilities , that is, probabilities of the occurrence of 0, 1,00, 01, 10, 11, 000, etc – in short, the probabilities of occurrence of all possiblebinary words. This means that the CA rule treated as a map in the space ofprobability measures is an infinitely-dimensional map .Infinite-dimensional maps are difficult to investigate, even numerically, thusfrom the early days of CA research, efforts were made to find a way to approxi-mate them by finite-dimensional maps. In a seminal paper [1], published over 30 a r X i v : . [ n li n . C G ] F e b ears ago, H. Gutowitz et al. proposed such an approximation, which they calledthe local structure theory . It was an application of a well know idea of Bayesianextension, widely used in statistical physics as a basis of so-called mean-fieldtheories, finite-cluster approximations, and related methods.Since 1987 the local structure theory has been widely used in CA research,as witnessed by a large number of citations of [1]. This could be somewhatsurprising, given that relatively few rigorous results are known about the localstructure theory. Usually, the authors using this method simply construct afinite-dimensional map or recurrence equations following the recipe given in [1],and declare that these posses orbits approximating the dynamics of the actualCA or related system which they investigate. Judgments on the quality of theapproximation are usually made based on numerical iterations of local structuremaps and numerical simulations of the CA in question. Numerical results arethus compared with other numerical results.In recent years, however, partial orbits of Bernoulli measures have been com-puted for some selected elementary CA [2], making a somewhat more rigorousapproach possible. The goal of this paper is to provide an example of a CArule for which some block probabilities are known exactly, and for which localstructure equations can be analyzed rigorously, without relying exclusively onnumerical iterations. This way, the quality of the approximation could be eval-uated in a solid and rigorous fashion, without worrying about numerical errors,finite size effects, etc.We selected elementary CA rule 14 as the most promising example for suchstudy. It has several interesting features: exact probabilities of blocks of lengthup to three are known for the orbit of the symmetric Bernoulli measure underthis rule, and some of these block probabilities exhibit non-trivial behaviour -for example, convergence toward the steady state as a power law with fractionalexponent. At the same time, rule 14 conserves the number of pairs 10 [3], andthe existence of this additive invariant provides a constrain simplifying localstructure equations, making them easier to analyze. Since block probabilities oflength 3 are known for this rule, we will construct local approximation of level 3and investigate its dynamics not only by simple numerical iterations, but byfinding invariant manifolds at the fixed point and determining the nature of theflow on these manifolds.One should stress here that in what follows we will use only very minimalformalism. More formal details about the construction of probability measuresover infinite bisequences and the construction of local structure maps for arbi-trary rules (both deterministic and probabilistic) can be found in [4], where thereader will also find more references on these subjects. Preliminary remarks about rule 14
Consider the fully discrete dynamical system (called cellular automaton ) where s i ( n ) ∈ { , } is the state of site i ∈ Z at time n ∈ N , with dynamics definedby s i ( n + 1) = f ( s i − ( n ) , s i ( n ) , s i +1 ( n )) . The function f : { , } → { , } s called the local rule . In this paper, we will consider f which is defined by f ( x , x , x ) = x + x − x x − x x − x x + x x x , and we call the above rule 14 , following the numbering scheme of Wolfram [5].Usually, the initial state at n = 0 is drawn from the Bernoulli distribution,where each site s i (0) is either in state 1 with probability ρ , or in state 0 withprobability 1 − ρ , independently of each other, where ρ ∈ [0 , ρ = 1 / a in a configuration obtained after n iterations of the rule, assuming that the initial configuration is drawn from theBernoulli distribution. Such probability will be denoted by P n ( a ) and called blockprobability . It is easy to show that if the initial distribution is Bernoulli, then theprobability of occurrence of a is independent of its position in the configuration.We will call such block probabilities shift invariant .The set of shift-invariant block probabilities P n ( a ) for all binary strings a defines a shift-invariant probability measure on the set of infinite binary bise-quences, but we will not be concerned with the formal construction of such mea-sures here. Interested reader can find all relevant details and references in [4].Consider now a configuration in which s i ( n + 1) = 1. By using the def-inition of rule f , one can easily figure out that s i ( n + 1) is determined en-tirely by the triple ( s i − ( n ) , s i ( n ) , s i +1 ( n )), and that the only possible values of( s i − ( n ) , s i ( n ) , s i +1 ( n )) producing s i ( n + 1) = 1 are (0 , , , ,
0) or (0 , , n + 1 is equal to the sumof probabilities of ocurrence of blocks 001, 010, and 011 at time n , P n +1 (1) = P n (001) + P n (010) + P n (011) . One can carry out a similar reasoning for longerblocks. For example, a pair of 1s, that is, s i ( n + 1) = 1 and s i +1 ( n + 1) = 1, canappear only and only if at the previous time step n the lattice positions i − , i, i +1 , i + 2 assumed values 0,0,1,0 or 0,0,1,1, i.e., ( s i − ( n ) , s i ( n ) , s i +1 ( n ) , s i +2 ( n )) =(0 , , ,
0) or ( s i − ( n ) , s i ( n ) , s i +1 ( n ) , s i +2 ( n )) = (0 , , , P n +1 (11) = P n (0010) + P n (0011) . Obviously, one can write analogous equations for probabilities of any binaryblock, obtaining an infinite system of difference equations. The complete set ofsuch equations for blocks of length up to 3 for rule 14 is shown below. P n +1 (0) = P n (000) + P n (100) + P n (101) + P n (110) + P n (111) ,P n +1 (1) = P n (001) + P n (010) + P n (011) ,P n +1 (11) = P n (0010) + P n (0011) ,P n +1 (00) = P n (0000) + P n (1000) + P n (1100) + P n (1101) + P n (1110) + P n (1111) ,P n +1 (01) = P n (0001) + P n (1001) + P n (1010) + P n (1011) ,P n +1 (10) = P n (0100) + P n (0101) + P n (0110) + P n (0111) ,P n +1 (000) = P n (00000) + P n (10000) + P n (11000) + P n (11100) + P n (11101)+ P n (11110) + P n (11111) ,P n +1 (001) = P n (00001) + P n (10001) + P n (11001) + P n (11010) + P n (11011) ,P n +1 (010) = P n (10100) + P n (10101) + P n (10110) + P n (10111) , n +1 (011) = P n (00010) + P n (00011) + P n (10010) + P n (10011) ,P n +1 (100) = P n (01000) + P n (01100) + P n (01101) + P n (01110) + P n (01111) ,P n +1 (101) = P n (01001) + P n (01010) + P n (01011) ,P n +1 (110) = P n (00100) + P n (00101) + P n (00110) + P n (00111) ,P n +1 (111) = 0 . (1)One thing which is immediately obvious is that not all of these equations are in-dependent because the block probabilities themselves are not independent. Blockprobabilities must satisfy so-called Kolmogorov consistency conditions , which arein fact just additivity conditions satisfied by a measure induced by block proba-bilities. For example, we must have P n (1) + P n (0) = 1, P n (01) + P n (00) = P n (0),etc. Consistency conditions can be used to express some block probabilities byothers. One can show that for binary strings, among probabilities of blocks oflength k , only 2 k − are independent [4], in the sense that one can choose 2 k − block probabilities which are not linked to each other via consistency conditions.For blocks of length up to 3, there are 14 block probabilities, P n (0), P n (1), P n (00), P n (01), P n (10), P n (11) P n (000), P n (001), P n (010), P n (011), P n (100), P n (101), P n (110), and P n (111). Among them only 2 − = 4 are independent.While there is some freedom in choosing which ones are to be treated as inde-pendent, we will choose the following four, P n (0), P n (00), P n (000), and P n (010).This is called the short block representation , and a detailed algorithm for choos-ing block this way is described in [4]. Here it is sufficient to say that short blockrepresentation ensures that the blocks selected as independent are the shortestpossible ones.Using consistency conditions, one can now express the remaining blocks oflength up to 3 in terms of P n (0), P n (00), P n (000), and P n (010), as follows: P n (1) = 1 − P n (0) ,P n (01) = P n (0) − P n (00) ,P n (10) = P n (0) − P n (00) ,P n (11) = 1 − P n (0) + P n (00) ,P n (001) = P n (00) − P n (000) ,P n (011) = P n (0) − P n (00) − P n (010) ,P n (100) = P n (00) − P n (000) ,P n (101) = P n (0) − P n (00) + P n (000) ,P n (110) = P n (0) − P n (00) − P n (010) ,P n (111) = 1 − P n (0) + 2 P n (00) + P n (010) . (2)Using the above substitutions one can reduce eqs. (1) to the following set of fourequations, P n +1 (0) = 1 − P n (0) + P n (000) , (3) P n +1 (00) = 1 − P n (0) + P n (00) + P n (000) , n +1 (000) = 1 − P n (0) + 2 P n (00) + P n (000) + P n (010) − P n (01000) ,P n +1 (010) = P n (0) − P n (00) + P n (000) . Note that the above cannot be iterated, because on the right hand side, in addi-tion to the four aforementioned independent probabilities, we have probability P n (01000), the probability of the block of length 5.Fortunately, in spite of the above problem, if the initial Bernoulli measureis symmetric, exact expressions for probabilities P n (0), P n (00), P n (000) and P n (010) for rule 14 (that is, the solution of eqs. (3)) can be obtained by combi-natorial methods. We will quote the relevant results below, omitting the proof,which can be found in [6]. Proposition 1 (Fuk´s et al. 2009).
For elementary rule 14, if the initial con-figuration is drawn from symmetric Bernoulli distribution, the probabilities ofblock of length up to 3 are given by P n (0) = 12 (cid:18) n − n C n − (cid:19) , (4) P n (00) = 2 − − n ( n + 1) C n + 14 , (5) P n (000) = 2 − n − (4 n + 3) C n , (6) P n (010) = 2 − − n ( n + 1) C n , (7) where C n is the n -th Catalan number, C n = n +1 (cid:0) nn (cid:1) . Note that although the above proposition provides probabilities of P n (0), P n (00), P n (000) and P n (010) only, the remaining probabilities of blocks of length up to3 can be easily computed using eqs. (2).Although we know exact solution of eqs. (3), we can also attempt to obtainan approximate solution by approximating the “problematic” block probabil-ity P n (01000). There exists a method for approximating longer block proba-bilities by probabilities of shorter blocks. This method is called the Bayesianextension , and it is known to produce block probabilities satisfying consistencyconditions [4]. Applying the Bayesian extension to P n (01000), one obtains P n (01000) ≈ P n (010) P n (100) P n (000) P n (10) P n (00) . (8)In the above, by definition, the fraction on the right hand side is considered tobe zero whenever its denominator is equal to zero. Using eqs. (2) we can nowexpress P n (01000) in terms of our four independent block probabilities, P n (01000) ≈ P n (010) ( P n (00) − P n (000)) P n (000)( P n (0) − P n (00)) P n (00) . (9)If we replace P n (01000) in eqs. (3) by the above approximation, we will obtainthe system of four coupled difference equations, x n +1 = − x n + z n + 1 , (10) n +1 = − x n + y n + z n + 1 , (11) z n +1 =1 + z n + v n − x n + 2 y n − v n ( y n − z n ) z n y n ( x n − y n ) , (12) v n +1 = x n − y n + z n , (13)where for brevity we introduced variables x n = P n (0), y n = P n (00), z n = P n (000). and v n = P n (010). Equations (10)–(13) will be referred to as localstructure equations of level 3, following nomenclature of [1,4]. The designation“level 3” pertains to the fact that we used block probabilities of length up to 3. How does the orbit of local structure equations (10)–(13) compare with knownexact solutions given by eq. (4)–(7)? In order to find this out, we will assumethat the initial probability measure is symmetric Bernoulli, meaning that x = P (0) = 1 / y = P (00) = 1 / z = P (000) = 1 /
8, and v = P (010) = 1 / Fig. 1.
Differences between exact and approximate values of block probabilities as afunction of n . Two differences are shown, P n (00) − y n (lower curve) and P n (000) − z n (upper curve). (7) and values obtained by iterating local structure equations (10)–(13). Twoifferences are shown, P n (00) − y n (lower curve) and P n (000) − z n (upper curve).In both cases we can see that the difference tends to zero as n → ∞ . Values of P n (0) − x n and P n (010) − v n (not shown) exhibit similar behaviour.This indicates that even though the local structure approximation of level3 does not produce exact values of block probabilities at finite n , it seems tobecome exact in the limit n → ∞ . To verify this, let us first note that from eq.(4)–(7) we obtainlim n →∞ P n (0) = 12 , lim n →∞ P n (00) = 14 , lim n →∞ P n (000) = 0 , lim n →∞ P n (010) = 0 . We will denote these values by ( x (cid:63) , y (cid:63) , z (cid:63) , v (cid:63) ) = ( , , , x (cid:63) , y (cid:63) , z (cid:63) , v (cid:63) ) is a fixed point of eqs. (10)–(13). In what follows, we willinvestigate stability of this fixed point. We will prove that the following propertyholds. Proposition 2.
If the dynamical system given by eqs. (10)–(13) is iteratedstarting from initial conditions x = 1 / , y = 1 / , z = 1 / , and v = 1 / ,then lim n →∞ ( x n , y n , z n , v n ) = ( x (cid:63) , y (cid:63) , z (cid:63) , v (cid:63) ) = (cid:0) , , , (cid:1) . This means that the local structure map approximates the exact probabilitiesremarkably well, converging to the same fixed point as the exact values. We willprove Proposition 2 by reducing local structure equations to two dimensions andby computing local manifolds at the fixed point.
Reduction to two dimensions
Close examination of equations (10)–(13) reveals some obvious symmetries. Firstof all, it is easy to check that x n +1 − y n +1 = x n − y n . Since x − y = , we have x n − y n = for all n , thus x n = y n + 14 . (14)Further simplification is possible. Note that v n +1 − y n +1 = 3( x n − y n ) − · − − . This implies that for any n > v n +1 = y n +1 − , or, equivalently,that for any n > v n = y n − . (15)Note that this does not hold for n = 0, because in this case v = y − /
8. Now,using eqs. (14) and (15), we can reduce our dynamical system to two dimension,as eqs. (11) and (12) become y n +1 = − y n + 14 ) + y n + z n + 1 ,z n +1 =1 + z n + ( y n −
14 ) − y n + 14 ) + 2 y n − ( y n − ) ( y n − z n ) z n y n (cid:0) ( y n + ) − y n (cid:1) . fter simplification we obtain, for n ≥ y n +1 = 12 − y n + z n , (16) z n +1 = z n − (4 y n −
1) ( y n − z n ) z n y n , (17)where we start the recursion at n = 1, taking y = 3 / z = 7 /
32. The last twovalues were obtained by direct computation of y and z from eqs. (11) and (12)for n = 0, by substituting x = 1 / y = , z = v = 1 / Proposition 3.
If the dynamical system described by eqs. (16) and (17) is it-erated starting at y = 3 / , z = 7 / , then lim n →∞ ( y n , z n ) = (cid:18) , (cid:19) . (18)In order to prove the above proposition let us first denote x = (cid:20) yz (cid:21) . In thisnotation, eqs. (16) and (17) define two-dimensional map F ( x ) = (cid:34) − y + zz − (4 y − y − z ) zy (cid:35) . (19)It is easy to check that the map F has the fixed point x (cid:63) = (cid:20) (cid:21) . In order toprove Proposition 3, all we need is to show that x (cid:63) is asymptotically stable (orat least semi-stable in the relevant domain).The Jacobian matrix of F evaluated at the fixed point x (cid:63) is given by A = (cid:20) − (cid:21) , and its eigenvalues are − x ∗ is a non-hyperbolic fixed point and one cannotdetermine its stability by eigenvalues alone. We will investigate its stability byresorting to the center manifold theory.Let P be the matrix of column eigenvectors of A , and let P − be its inverse, P = (cid:20) (cid:21) , P − = (cid:20) − (cid:21) . We will first move the fixed point to the origin and simultaneously diagonalizethe linear part of F . The following change of variables accomplishes this task, X = P − ( x − x (cid:63) ) , (20)here the components of the new variable X will be denoted by Y and Z . Eq.(20) thus yields Y = y − − z , (21) Z = z. (22)Change of variables from x to X transforms the dynamical system x n +1 = F ( x n )into the system X n +1 = P − F ( P X n + x (cid:63) ) − P − x (cid:63) . (23)This yields, after simplification, Y n +1 = − Y n + 12 (4 Y n + 2 Z n ) (cid:0) Y n − Z n + (cid:1) Z n Y n + Z n + , (24) Z n +1 = Z n − (4 Y n + 2 Z n ) (cid:0) Y n − Z n + (cid:1) Z n Y n + Z n + . (25)One can immediately see that the above system has (0 ,
0) as a fixed point, andthat its linear part is given by Y n +1 = − Y n , Z n +1 = Z n . As mentioned earlier,there is nothing we can say about the stability of (0 ,
0) by examining the linearpart alone, except that in the vicinity of (0 ,
0) the Y variable is changing its signat each iteration. We will use the method outlined in [7] to find the invariantmanifold corresponding to − flipmanifold and denote it by W f .Let us assume that W f has the equation Z = h ( Y ), where h in the vicinityof 0 is given by the series h ( Y ) = a Y + a Y + a Y + a Y + . . . . Note that theseries starts from the quadratic term, and this is because the manifold Z = h ( Y )must be tangent to the Y axis (we already diagonalized our dynamical system).The condition for invariance of W f requires that the relationship Z n = h ( Y n )remains valid in the next time step, meaning that Z n +1 = h ( Y n +1 ). Let us rewriteeqs. (24) and (25) as Y n +1 = G ( Y n , Z n ) , (26) Z n +1 = G ( Y n , Z n ) , (27)where G ( Y, Z ) = − Y + 12 (4 Y + 2 Z ) (cid:0) Y − Z + (cid:1) ZY + Z + , (28) G ( Y, Z ) = Z − (4 Y + 2 Z ) (cid:0) Y − Z + (cid:1) ZY + Z + . (29)Condition Z n +1 = h ( Y n +1 ) now becomes G ( Y, Z ) = h ( G ( Y, Z )) , and, by taking Z = h ( Y ), it yields G ( Y, h ( Y )) = h ( G ( Y, h ( Y ))) . (30)his means that if we expand G ( Y, h ( Y )) − h ( G ( Y, h ( Y ))) into the Taylorseries with respect to Y , all coefficient of the expansion should be zero. Suchexpansion, done by the Maple symbolic algebra system, yields G ( Y, h ( Y )) − h ( G ( Y, h ( Y ))) = (2 a − a ) Y + (cid:18) − a − (cid:18) a (cid:19) a + 16 a + 4 a (cid:19) Y + (cid:18) a − a − (cid:18) a (cid:19) a − a a + 16 a + 16 a − (cid:18) − a − (cid:18) a (cid:19) a + 16 a (cid:19) a (cid:19) Y + O (cid:0) Y (cid:1) . Coefficients in front of Y , Y , Y , . . . must be zero, yielding the system of equa-tions for a , a , a , . . . ,0 = 2 a − a , − a − (cid:18) a (cid:19) a + 16 a + 4 a , a − a − (cid:18) a (cid:19) a − a a + 16 a + 16 a (31) − (cid:18) − a − (cid:18) a (cid:19) a + 16 a (cid:19) a ,. . . Solving the above system one obtains a = 4, a = 8, a = 3, a = −
32, etc.The flip manifold W f is, therefore, given by Z = h ( Y ) = 4 Y + 8 Y + 32 Y − Y + O ( Y ) . (32)By substituting Z n by h ( Y n ) on the right hand side of eq. (24) and Taylorexpanding again one obtains the equation describing the dynamics on the flipmanifold W f , Y n +1 = − Y n + 8 Y n + 32 Y n + O (cid:0) Y n (cid:1) . (33)The above equation has 0 as a fixed point, and we need to determine its stability.Recall that a fixed point ¯ x of x n +1 = f ( x ) is said to be asymptotically stable ifthere exist δ > x satisfying | x − ¯ x | < δ we have lim n →∞ x n =¯ x . We will use the following general test for asymptotic stability [8]. Theorem 1 (Murakami 2005).
Let ¯ x be a fixed point of x n +1 = f ( x n ) . Sup-pose that f ∈ C k − ( R ) , f (cid:48) (¯ x ) = − , f j (¯ x ) = 0 for j ∈ { , , . . . , k − } , andthat f ( k ) (¯ x ) (cid:54) = 0 . If k is odd and f ( k ) (¯ x ) > , then ¯ x is asymptotically stable. In our case, for eq. (33), f ( x ) = − x + 8 x + 32 x + O (cid:0) x (cid:1) , ¯ x = 0, f (cid:48) (¯ x ) = − f (cid:48)(cid:48) (¯ x ) = 0, and f (3) (¯ x ) = 48, thus the theorem applies, meaning that zero isasymptotically stable fixed point of eq. (33). c W f W c W f Fig. 2.
The flip manifold W f and the center manifold W c in transformed ( Y, Z ) co-ordinates (top) and original ( y, z ) coordinates (bottom). Points represent numericallycomputed orbits of a sample point on W c ( ◦ ) and W f ( • ). We need to perform a similar analysis for the eigenvalue 1 and the correspond-ing center manifold W c . Let us assume that W c has equation Y = g ( Z ), where g in the vicinity of 0 is given by the series g ( Z ) = b Z + b Z + b Z + b Z + . . . .The condition for invariance of W c requires that Y n = g ( Z n ) remains valid atthe next time step, Y n +1 = g ( Z n +1 ). Using our previous notation this meansthat G ( Y, Z ) = g ( G ( Y, Z )), which, by substituting Y = g ( Z ), yields G ( g ( Z ) , Z ) = g ( G ( g ( Z ) , Z )) . (34)As before, by expanding G ( g ( Z ) , Z ) = g ( G ( g ( Z ) , Z )) into the Taylor seriesand setting all coefficient of the expansion to be zero we obtain, using Maple, b = , b = − , b = − , b = − , etc. The equation of the center manifold is,herefore, Y = 12 Z − Z − Z − Z + O (cid:0) Z (cid:1) . (35)By substituting Y n by g ( Z n ) on the right hand side of eq. (25) and Taylorexpanding again one obtains the equation describing the dynamics on the centermanifold W c , Z n +1 = Z n − Z n + 6 Z n − Z n + O (cid:0) Z n (cid:1) . (36)In order to determine the stability of 0 in the above difference equation,let us first define semistability. A fixed point ¯ x of x n +1 = f ( x ) is said to be asymptotically semistable from the right if there exist δ > x satisfying x − ¯ x < δ we have lim n →∞ x n = ¯ x . One can show [9] that if f (cid:48) (¯ x ) = 1and f (cid:48)(cid:48) (¯ x ) < x is assymptotically stable from the right. In our case, foreq. (36), we have f ( x ) = x − x + 6 x − x + O (cid:0) x (cid:1) , ¯ x = 0, f (cid:48) (¯ x ) = 1 and f (cid:48)(cid:48) (¯ x ) = − <
0, thus for eq. (36), zero is asymptotically semistable from theright.Figure 2 shows manifolds W f and W c together with sample orbits generatednumerically by iterating eqs. (24) and (25). Direction of the flow is indicated byarrows. Note that W c is asymptotically semistable only on the right (for Z >
Z < Z represents the probability of 000 block, thus it must always be positive.Since 0 is asymptotically stable on W f , and asymptotically semistable on W c , we conclude that for Z >
0, lim n →∞ ( X n , Z n ) = (0 , n →∞ ( x n , z n ) = (1 / , (cid:50) We have demonstrated so far that for rule 14, the local structure approximationof level 3 reproduces correctly the limiting values of probabilities of blocks oflength up to 3. What about the rate of convergence to these limiting values?In order to find this out, let us consider rates of convergence to zero of P n (000)and its approximation z n . We know that P n (000) = 2 − n − (4 n + 3) C n , where C n = n +1 (cid:0) nn (cid:1) = (2 n )! n !( n +1)! . Using Stirling’s formula for large n , n ! ∼ √ πn (cid:16) ne (cid:17) n , the Catalan number C n can be approximated as C n = 1 n + 1 (2 n )!( n !) ∼ n + 1 √ πn (cid:0) ne (cid:1) n (cid:0) √ πn (cid:0) ne (cid:1) n (cid:1) = 1 n + 1 2 n √ πn = 1 n + 1 4 n √ πn , meaning that P n (000) converges toward zero as a power law P n (000) ∝ n − / ,where x ∝ y means the ratio x/y tends to a positive number as n → ∞ .Let us now examine convergence of z n to 0. We do not have a formula for z n ,but we can generate z n numerically, by iterating the local structure equations.Figure 3 shows the graph of z n vs. n in log-log coordinates together with theraph of P n (000) vs. n . We can see that both graphs appear to be almost straightlines, confirming that both z n and P n (000) behave as n α for large n . The differ-ence is in the value of the exponent α . For P n (000) the exponent (computed asa slope of the upper line in Figure 3) is α ≈ − /
2, whereas for z n the exponent(computed as a slope of the lower line) is α ≈ − Fig. 3.
Plot of P n (000) (upper line) and its local structure approximation z n (lowerline) as a function of n in log-log coordinates. The value of the exponent α ≈ − y = 3 / z = 7 /
32, liesalmost on the center manifold W c . The convergence toward the fixed point is,therefore, dominated by eq. (36), which, if we keep only leading terms, becomes Z n = Z n − Z n . Although this equation is not solvable in a closed form, we canobtain its asymptotic solution using the standard technique used in the theoryof iterations of complex analytic functions. We can namely conjugate the map Z → Z − Z with appropriate M¨obius transformation, which moves the fixedpoint to ∞ [10,11]. In our case, the M¨obius map will simply be the inverse,meaning that we change variables in the equation Z n = Z n − Z n to u n = 1 /Z n ,obtaining u n +1 = u n + 2 + 4 u n − . (37)Since u n → ∞ , the above can be approximated for large n by u n +1 = u n + 2,which has the solution u n = 2 t + u , or, going back to the original variable, Z n = t +1 /z . The result z n = Z n ∝ t − immediately follows.In conclusion, one could thus say that the local structure approximation cor-rectly reproduces not only the coordinates of the the fixed point but also the ype of convergence toward the fixed point (as a power law). It fails, however,to reproduce the correct value of the exponent in the power law. This in agree-ment with the commonly reported results of investigations of critical phenom-ena: mean-field type theories cannot reproduce values of fractional exponents inpower laws.It would be interesting and beneficial to extend results of this paper to non-symmetric initial Bernoulli measures. Numerical evidence suggests that localstructure approximation remains exact in the limit of n → ∞ in such cases, butto be sure one would need to generalize eqs. (4)–(7) to non-symmetric initialmeasure. This, in principle, should be possible, and will be attempted in thefuture. Acknowledgement:
H.F. acknowledges financial support from the Natural Sciences and Engineer-ing Research Council of Canada (NSERC) in the form of Discovery Grant.
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