Infinite Number of Disjoint Chaotic Subsystems of Cellular Automaton Rule 106

In this paper, the dynamics of rule 106, a Chua’s hyper Bernoulli cellular automata rule, is studied and discussed from the viewpoint of symbolic dynamics. It is presented that rule 106 defines a chaotic subsystem which is topologically mixing and possesses the positive topologically entropy. An effective method of constructing its chaotic subsystems is proposed. Indeed, it is interesting to find that this rule is filled with infinitely many disjoint chaotic subsystems. Special attention is paid to each subsystem on which rule 106 is topologically mixing and possesses the positive topologically entropy. Therefore, it is natural to argue that the intrinsic complexity of rule 106 is high from this viewpoint.


Introduction
Cellular Automata (CA), first conceived around 1950 by von Neumann [1], are a class of spatially and temporally discrete mathematical structure by local interactions and an inherently parallel form of evolution.The whole structure is able to produce complex and interesting dynamical phenomena by means of designing simple transition rule.Due to their simple mathematical constructions and distinguishing features, CA have drawn a great deal of attention from various scientists.In 1969, the study of topological dynamics of CA was developed by Hedlund [2], who viewed one-dimensional CA in the context of symbolic dynamics as endomorphisms of the shift dynamical system, where the main results are the characterizations of surjective and open CA.In 1970, Conway proposed game of life [3], which received widespread interests among researchers in different fields.In the early 1980s, Wolfram proposed CA as models for physical systems exhibiting complex or even chaos behaviors and elementary CA (ECA) that consist of a one-dimensional array of finite binary cells, each interacting only with the two nearest neighbors [4]- [6].He classified 256 ECA rules informally into four classes using dynamical concepts like periodicity, stability and chaos.In 2002, Wolfram introduced his work A New Kind of Science [6].Based on this work, Chua et al. have concluded the dynamics of ECA from a nonlinear dynamics perspective [7]- [10].And he divided 256 ECA rules into four classes: periodk rules ( ) , Bernoulli-shift rules, complex Bernoulli-shift rules and hyper Bernoulli-shift rules.Gratefully, the research of CA has drawn more and more scientists' attention in the last 20 years.Many concepts of topological dynamics have been used to describe and classify them [11]- [15].And the dynamical properties of some robust Bernoulli-shift rules have been studied in the bi-infinite symbolic sequence space [14], [15].Rule 106 belonging to hyper Bernoulli-shift rules possesses complex and distinctive dynamical behaviors.In a paper [16], the authors introduced the notion of permutivity of a map in a certain variable.Then they proved that every one-dimensional CA based on the local rule which is permutive either in the leftmost or rightmost variable is Devaney chaotic.Rule 106 is in this situation.Presently, this work is devoted to an in-depth study of rule 106 from the perspective of nonlinear dynamics under the framework of bi-infinite symbolic sequence space, and mainly studies the complex dynamics on its infinite number of subsystems.
The rest of the paper is organized as follows: Section 2 presents the basic concepts of one-dimensional CA and symbolic dynamics.Based on these concepts, it shows a subsystem of rule 106.Section 3 explores the complex dynamical behaviors of rule 106.Section 4 describes that there exist infinitely many disjoint chaotic subsystems in this chaotic subsystem.Finally, Section 5 concludes this paper.

Preliminaries
For a finite symbol S , a word over S is finite sequence ( ) . For any CA there exists radius 0 r ≥ and a loca rule . Moreover, ( ) S T is a com- pact dynamical system.To enhance readability, it is desirable to write a CA as N T for local rule N .
A set and T -invariant, then ( ) , X T or simply X is called a subsystem of T .For instance, let A denote a set of some finite words over S , and Λ A is the set which consists of the bi-infinite configurations made up of all the words in A .Then Λ A is subsystem of σ , where A is said to be the determinative block system of Λ A .
Thus the global map of rule 106 is induced as follows: for any ( )  denotes the i th symbol of ( )

106
T x .For clarity, the truth table of rule 106 is depicted in Table 1.
Proof: (Necessity) Suppose that there exists a subset , According to the Boolean function of rule 106, one has (Sufficiency) The proof of sufficiency can be verified directly, the details are omitted here.The proof of the proposition is completed.
For illustration, simulations of the spatial and temporal evolution of rule 106 with a random initial configuration and an initial configuration of Λ are shown in Figure 1, where the black pixel stands for 1 and white for 0.

Complex Dynamics of T 106
In this section, the dynamical behaviors of 106 T on Λ are exploited.As the topological dynamics of a subshift of finite type is largely determined by the properties of its transition matrix, it is helpful to briefly review some definitions from [17].A matrix A is positive if all of its entries are nonnegative; irreducible if , , ( ) , , then the associated transition matrix A is the N N × matrix with , 1 Denote a 2-order subshift of finite type by : Then, the 2-order subshift Λ D of σ is defined by ( ) . Moreover, it is clear to see that the transition matrix A of the subshift Λ D is: are topologically conjugate; 2) 106 T Λ is topologically mixing; 3) the topological entropy of 106 T Λ satisfies : , , , , , , , , Then, it follows from the definition of Λ D that for any x ∈ Λ A , one has ( ) Then, it is easily to check that π is a homeomorphism and π σ σ π =   .
2) A satisfies 0, 4 n A n > ∀ ≥ ; namely, A is irreducible and aperiodic, which implies that σ is topologically mixing on Λ D .Then, one can deduce 106 T Λ is topologically mixing according to Theorem 1 1) and Proposition 1.
3) As It is noted that a positive topological entropy is an important signature of the complexity of the system.It follows from [18] that the positive topological entropy implies chaos in the sense of Li-Yorke.And the topologically mixing is a very complex property of dynamical systems.A system with topologically mixing property has many chaotic properties in different senses.Therefore, the above mathematical analysis provides the following result.
Theorem 2. 1) 106 T is chaotic in the sense of Li-Yorke; 2) 106 T is chaotic in the sense of both Li-Yorke and Devaney on Λ .

Infinitely Many Chaotic Subsystems of T 106 in Λ
It is helpful to review some definitions and basic properties of releasing transformation before we discuss the dynamics of 106 T on infinite number of subsystems.Let . Denote by Z S  the space of bi-infinite configurations over S  and induce a matric " d " onto Z S  as defined in the preceding section.Then, the releasing transformation R is defined as follows: : , if mod 0, and ; , if 1 mod 0, and ; , if 1 mod 0, and ; , if mod 0, and ; , if 1 mod 0, and ; , if 1 mod 0, and .

Proposition 2. [19]
Releasing transformation R is a continuous and injective map.Let ( ) 2 0,1, 0,1, , 0,1, 0,1 , 0, 0, 0, , 0, 0, 0, 0 be a new symbolic set.Denote by i Λ  the subshift in Z S  determined by the transition matrix as 1 1 , where σ is the classical left-shift map.And let i Λ be ( )     x x = .So 106 T is injective.Since i Λ is a compact Hausdorff space, 106 i T Λ is one to one, onto and continuous map.T on i ′ Λ approaches 0 as i approaches +∞ .Meanwhile, it has been proved that there exists a "big" subsystem of rule 106, including infinite disjoint chaotic subsystems 1 2 , , ′ ′ Λ Λ  .This analytical assertion provides an enlightening fact that the hyper Bernoulli-shift rule 106 is full of infinite "small" chaotic subsystems in a "big" subsystem, demonstrating its very rich and complex dynamics.

Conclusion
One of the main challenges is to explore the quantitative dynamics in cellular automata evolution.Hyper

Figure 1 .
Figure 1.(a) The evolution of rule 106 from random initial configuration, (b) The evolution of rule 106 from an initial configuration of Λ .
elements in A , respectively.Then one can construct a new symbolic space ˆZ S on Ŝ .Denote by radius of the transition matrix A .Proof: 1) Define a map from Λ to Λ D as follows:

2 )Remark 1 .
→ Λ exists and continuous.Therefore, 106 : is also an open set, thus, one has It is easily deduced by Theorem 3 (2) and Theorem 4 (each i Z + ∈ , i ′ Λ is closed and σ -invariant.Thus Theorem 3 and 4 also hold for i ′ Λ , where i Z + ∈ .It is important to point out that the topologically entropy of 106

Table 1 .
Logical table of rule 106.
[18]r subshift of finite type is topologically mixing if and only if its transition matrix is irreducible and aperiodic[17][18].The nonlinear dynamical behavior of 106 T on Λ is discussed by establishing the topologically conjugate relationship between ( ) It is clear to check that A  is irreducible and aperiodic, thus σ is topologically mixing on i the following are two conditions to illustrate: i