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Journal of Mathematics
Volume 2013 (2013), Article ID 404626, 6 pages
The Measure-Theoretic Entropy and Topological Entropy of Actions over
1Department of Applied Mathematics, Feng Chia University, Taichung 40724, Taiwan
2Taipei Municipal Minsheng Junior High School, Taipei 10591, Taiwan
Received 31 January 2013; Accepted 29 May 2013
Academic Editor: Mike Tsionas
Copyright © 2013 Chih-Hung Chang and Yu-Wen Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper studies the quantitative behavior of a class of one-dimensional cellular automata, named weakly permutive cellular automata, acting on the space of all doubly infinite sequences with values in a finite ring , . We calculate the measure-theoretic entropy and the topological entropy of weakly permutive cellular automata with respect to any invariant measure on the space . As an application, it is shown that the uniform Bernoulli measure is the unique maximal measure for linear cellular automata among the Markov measures.
Cellular automata (CA for brevity), introduced by Ulam and von Neumann, have been systematically studied by Hedlund from purely mathematical point of view . The study of such dynamics called CA has received remarkable attention in the last few years [1–3]. CA have been widely investigated in a lot of disciplines (e.g., mathematics, physics, and computer science). In , the dynamical behavior (ergodicity and topological transitivity) of -dimensional linear CA (LCA) over the ring has been studied. Some open questions concerning the topological and ergodic dynamics of one-dimensional CA have been addressed in .
It is well known that there are several notions of entropy (i.e., measure-theoretical, topological, and directional) of measure-preserving transformation on probability space in ergodic theory. It is important to know how these notions are related with each other. In the last years, a lot of works are devoted to this subject (see, e.g., [5–13]). Recall that by the Variational Principle the topological entropy is the supremum of the entropies of invariant measures. In [8, 14, 15], the authors showed that the uniform Bernoulli measure is a measure of maximal entropy for some one-dimensional CA.
The notion of entropy has been extensively studied in many disciplines (e.g., computer science, mathematics, physics, chemistry, and information theory) with different purposes. This notion first arose in thermodynamics as a measure of the heat absorbed (or emitted), when external work is done on a system. In probability theory, it constitutes a measure of the uncertainty. The entropy has been interpreted as a measure of the chaotic character of a dynamical system by many authors (see [4, 16, 17]), the value has been in general accepted as a measure of the complexity of the dynamics of over the space . Some authors have stated that the topological entropy of a map is a crude global measure of the exponential complexity of the structure of the orbits of the map (see [4, 17]). Badii and Politi  have studied the complexity exhibited by some CA with elementary rule by using both topological (graph-theoretical) and metric (thermodynamic) techniques. Lloyd and Pagels  have defined a measure complexity for the macroscopic states of physical systems. They have proved that the average complexity of a state must be proportional to Shannon entropy of the set of trajectories, .
In this paper we study the measure-theoretical entropy and topological entropy of one-dimensional weakly permutive CA (defined later) acting on the space of all doubly infinite sequences with values in a finite ring , . In , the author computed the measure-theoretical entropy with respect to the uniform Bernoulli measure for the case where is linear defined by the local rule . Recently, Ban et al.  studied the complexity of permutative CA (defined later) in thermodynamics and topological aspects, and they also gave the formulae to compute measure-theoretic and topological entropies. In this paper, for both measure-theoretic and topological entropies, we extend results obtained in [8, 14, 15, 20] to the case that is a weakly permutive CA over the ring with respect to any invariant measure. We remark that LCA is a special case of weakly permutive CA. In addition, the formula of topological entropy extends D’amico et al.’s result  for the topological entropy of LCA. We also show that the uniform Bernoulli measure is the unique maximal measure for LCA, whenever we focus on the Markov measures.
We give an example herein. Suppose that the local rule is given by (cf. Figure 1). The local rule is nonlinear. Applying Theorems 4 and 11, we conclude that the topological entropy of the prescribed weakly permutive CA is (see Example 10).
The rest of the elucidation is organized as follows. The upcoming section gives some definitions and results of permutive and weakly permutive CA. Section 3 demonstrates formulae of the measure-theoretic and topological entropies. The uniform Bernoulli measure is a measure of maximal entropy that is also demonstrated in Section 3.
Let , and let be the space of bi-infinite sequence . Hedlund examines CA in the viewpoint of symbolic dynamical systems . He shows that is a CA if and only if can be represented as a sliding block code; that is, there exist and a block map such that for and . Such is called the local rule of . The study of the local rule of a CA is essential for the understanding of this system.
A local rule is called leftmost (resp., rightmost) permutive if there exists an integer , (resp., ), such that (i)is a permutation at whenever the other variables are fixed; (ii) does not depend on for (resp., ). is called bipermutive provided is both left-most and right-most permutive. The family of permutive cellular automata consists of the following three types of local rules: (1) is leftmost permutive and does not depend on for ;(2) is rightmost permutive and does not depend on for ;(3) is bipermutive. Without loss of generality, we may assume that depends only on for and the coefficients of and are both nonzero in . Let be factorized into the product of prime factors. Suppose that a local rule is given, and set . is called weakly permutive if, for , there is such that is either constant, a multiple of an identity map, or permutive.
The present elucidation investigates the measure-theoretic and topological entropies of weakly permutive CA. For reader’s convenience, we recall definitions of measure-theoretic entropy, topological entropy, and topological pressures. Reader may refer to  for more details.
Let be an invariant probability measure on , and let and be two finite measurable partitions of . Define and by respectively. The measure-theoretic entropy of is defined by where the supremum is taken over all finite measurable partitions .
Define by It is easy to verify that is a metric and is a compact metric space. Moreover, let be a cylinder in , where . Then is not only open but close in . Let be an open cover of , denoted by where the infimum is taken over the set of finite subcovers of and denotes the cardinality of . The topological entropy of is defined by where the supremum is taken over all open covers .
Example 1. Let be defined as where is Gauss function. It is seen that is a permutation at but not permutive. Observe that is Garden of Eden for for some ; namely, there exists no preimage of . Then is not strongly mixing.
3. Entropy of Weakly Permutive Cellular Automata
Let be two compact topological spaces, and let be a probability measure on . If is onto, the push forward measure of on is defined by . It is wellknown that is also a probability measure.
Lemma 2. Let for some relative prime factors and . Denote Then provided , where and are the push forward measures of on and , respectively, and and are CA with local rules and , respectively.
Proof. Denote and . Define by
Observe that indicates that is an isomorphism. Moreover,
for . That is, the diagram
commutes. Thus is topologically conjugated to . The isomorphism indicates that .
The proof is completed.
Remark 3. Notably, the demonstration of Lemma 2 asserts that a CA defined on is topologically conjugated to the direct product of the projection of on and provided that and is relatively prime to .
Theorem 4 comes straightforwardly from the definition of weakly permutive CA and Lemma 2, and hence the proof is omitted.
Theorem 4. Let for some prime factors and . Suppose that is a weakly permutive CA, is -invariant, and , where is the push-forward measure of on for . Then
An immediate application of Theorem 4 is computing the measure-theoretic entropy of a linear CA (LCA). Herein is an LCA if its corresponding local rule is given by for some , .
Theorem 5. Let for some prime and . Suppose is an LCA and is -invariant such that . Then Moreover, suppose that is a Bernoulli measure. Denote for . Then where , .
Remark 6. Theorem 5 only presents the explicit form of the measure-theoretic entropy of an LCA with respect to a Bernoulli measure. The exact formula of the measure-theoretic entropy of an LCA with respect to a Markov measure can also be obtained. Since the formula is much complicated, we omit the case.
Before demonstrating Theorem 5, we introduce the following lemma.
Lemma 7. Let for some prime , . If is an LCA, then is weakly permutive.
Proof. Denote by the collection of linear local rules, and . Define by
It is easily seen that is bijective. Moreover, let denote the power series generated by over . Then defined by
is also a bijection. Observe that, for each ,
where . This implements that the diagram
commutes. Moreover, yielding the Mathematical Induction, we have for all , where .
Write as such that is consisting of those monomials whose coefficients are coprime to . We claim that, for all , It is seen that Suppose that In other words, for some . Therefore, This demonstrates our claim by the mathematical induction.
Let ; then , and is permutive. The proof is completed.
Theorem 8 (see ). Suppose that the local rule of a CA is given by , where , and is an -invariant Bernoulli measure. Denote and . One has the following results.(i)If is left permutive, then ; (ii)If is right permutive, then ; (iii)If is bipermutive, then .
In general, Lemma 7 does not hold. For instance, let be defined by . Then is weakly rightmost permutive. However, whose coefficient of is either or for all . Thus cannot be rightmost permutive for .
Example 9. Let , and let be the -Bernoulli measure. Then , where and . And the push forward measure of on and is respectively. Set ; then Theorem 5 indicates that
Example 10. Suppose . Let be given by . Then with , , and . It is seen that is a constant map, and and are both permutive. Hence is weakly permutive but not linear. Theorems 4 and 8 assert that
provided , where is defined in Theorem 5.
Similar to the discussion of the formula of the measure-theoretic entropy of weakly permutive CA, the topological entropy of weakly permutive CA can be obtained analogously.
Theorem 11. Let be factorized into the product of prime factors. Suppose that is a weakly permutive CA and is the projection of on for . Then
In , the authors demonstrated the formula of the topological entropy of LCA. Lemma 7 indicates that LCA is a proper subset of weakly permutive CA. That makes Theorem 11 an extension of Theorem 12. More precisely, (31) holds for the topological entropy of weakly permutive CA.
Theorem 12. Let be factorized into the product of prime factors. Suppose that is a LCA. Then where and are defined in (16).
Remark 13. The variational principle in thermodynamic formalism indicates that the topological entropy of a compact system is obtained by the supermum of its measure-theoretic entropies among all invariant measures, and a measure that attains the supremum is called a maximal measure. It comes immediately from Theorems 5 and 12 that, if we only consider invariant Markov measures, the uniform Bernoulli measure is the unique maximal measure for LCA.
Example 14. Let and be the same as in Example 9, and let be the uniform Bernoulli measure; then
Example 15. Let and be the same as in Example 10, and let be the uniform Bernoulli measure; then
Chih-Hung Chang is grateful for the partial support of the National Science Council, Taiwan (Contract no. NSC 101-2115-M-035-002-).
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