The Measure-Theoretic Entropy and Topological Entropy of Actions over Z m

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 Z m , m ≥ 2. We calculate the measuretheoretic entropy and the topological entropy of weakly permutive cellular automata with respect to any invariant measure on the space Z m . As an application, it is shown that the uniform Bernoulli measure is the unique maximal measure for linear cellular automata among the Markov measures.


Introduction
Cellular automata (CA for brevity), introduced by Ulam and von Neumann, have been systematically studied by Hedlund from purely mathematical point of view [1].The study of such dynamics called CA has received remarkable attention in the last few years [1][2][3].CA have been widely investigated in a lot of disciplines (e.g., mathematics, physics, and computer science).In [2], the dynamical behavior (ergodicity and topological transitivity) of -dimensional linear CA (LCA) over the ring Z  has been studied.Some open questions concerning the topological and ergodic dynamics of onedimensional CA have been addressed in [4].
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][6][7][8][9][10][11][12][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 ℎ top () 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 [18] have studied the complexity exhibited by some CA with elementary rule by using both topological (graph-theoretical) and metric (thermodynamic) techniques.Lloyd and Pagels [19] 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,  = − ∑    log   .
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 Z  ,  ≥ 2. In [14], 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. [15] 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 Z  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 [17] 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 4 log 3 + log 5 (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.

Preliminary
Let Z  = {0, 1, . . .,  − 1}, and let Ω = Z Z  be the space of bi-infinite sequence  = (  ) ∞ −∞ .Hedlund examines CA in the viewpoint of symbolic dynamical systems [1].He shows that  : Ω → Ω is a CA if and only if  can be represented as a sliding block code; that is, there exist  ∈ Z + and a block map  : Z 2+1  → Z  such that ()  = ( − , . . .,  + ) for  ∈ Ω and  ∈ Z.Such  is called the local rule of .The study of the local rule of a CA is essential for the understanding of this system.
is called bipermutive provided  is both left-most and rightmost 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  > 0; (2)  is rightmost permutive and does not depend on   for  < 0; (3)  is bipermutive.Without loss of generality, we may assume that ( ℓ , . . .,   ) depends only on   for ℓ ≤  ≤  and the coefficients of  ℓ and 2 ⋅     be factorized into the product of prime factors.Suppose that a local rule  is given, and set is either constant, a multiple of an identity map, or permutive.
The present elucidation investigates the measuretheoretic and topological entropies of weakly permutive CA.For reader's convenience, we recall definitions of measuretheoretic entropy, topological entropy, and topological pressures.Reader may refer to [13] for more details.
Let  be an invariant probability measure on (Ω, ), and let  and  be two finite measurable partitions of Ω. Define  ⋁  and   () by respectively.The measure-theoretic entropy of  is defined by where the supremum is taken over all finite measurable partitions .
where the infimum is taken over the set of finite subcovers P of P and # denotes the cardinality of .The topological entropy of  is defined by where the supremum is taken over all open covers P.
Example 1.Let  : Z 3 4 → Z 4 be defined as where [⋅] is Gauss function.It is seen that  is a permutation at  1 but not permutive.Observe that  is Garden of Eden for  ∈ { ∈ Z Z 4 :   = 0,  +1 = 1 for some  ∈ Z}; namely, there exists no preimage of .Then  is not strongly mixing.

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 for  ∈ Z.That is, the diagram commutes.Thus  is topologically conjugated to The proof is completed.
Remark 3. Notably, the demonstration of Lemma 2 asserts that a CA  defined on Z Z  is topologically conjugated to the direct product of the projection of  on Z Z  and Z Z  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.
for some prime factors   and   ∈ N. Suppose that  is a weakly permutive CA,  is -invariant, and  ≅  1 ×⋅ ⋅ ⋅×  , where   is the push-forward measure of  on Z Z

Theorem 5. Let 𝑚 = 𝑝
Moreover, suppose that  is a Bernoulli measure.Denote ) 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 measuretheoretic 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 ,  ∈ N. If  is an LCA, then  is weakly permutive.
Proof.Denote by L the collection of linear local rules, and It is easily seen that  is bijective.Moreover, let Z  [[,  −1 ]] denote the power series generated by {,  −1 } over Z  .Then is also a bijection.Observe that, for each b = (  ) ∈ Ω, where T ≡ ().This implements that the diagram commutes.Moreover, yielding the Mathematical Induction, we have   =  −1 (T  ) for all  ∈ N, where   =  ∘  −1 .Write T() as T() = T 1 () + T 2 () such that T 1 () is consisting of those monomials whose coefficients are coprime to .We claim that, for all  ∈ N, It is seen that Suppose that In other words, (T 1 () + T 2 ())   =  +1 () + T   1 () for some ().Therefore, This demonstrates our claim by the mathematical induction.

Theorem 11. Let 𝑚 = 𝑝
In [17], 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.
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.(33)

2 2
⋅    be factorized into the product of prime factors.Suppose that  is a weakly permutive CA and   is the projection of  on Z Z     for  = 1, . . ., .Then ℎ top () =  ∑ =1 ℎ top (  ) .