Topological Dynamics Of CA Rule 57

As a special dynamical system, Cellular Automata(CA) have simple mathematical structures and possess rich dynamical behaviors corresponding to the design of differ-ent rules. As the most simple kind of one-dimension CA, it can be considered as one-dimension symbolic dynamical system. In this paper we use the most well-known shift map to study elementary cellular automata(ECA), Especially, the dynamical behaviors of rule 57. this paper is organized as follows:Chapter 1 serves as an introduction. We review the context of CA briefly, introduce the historical development, as well as the two common classifications of ECA, in addition, CA have extensive applications in many scientific domain.Chapter 2 presents the basic concepts of elementary cellular automata and sym-bolic dynamics, and recalls the theorems, relative concepts, conclusions in topological dynamics.Chapter 3:Firstly, we discuss the basic concept of rule 57, show the Boolean function expression of rule 57, and introduce the topological conjugacy equivalence classification: rule 99. in particular, Two different homeomorphism maps g and h were given between the two rules. Then we do research on the relationships between injective, surjective and homeomorphism of CA using shift of finite type and properties of entropy, discuss injective and surjective between rule 57 and rule 99 specifically. The results showed that rule 57 and rule 99 are neither injective nor surjective. Moreover, in order to gain further insight into the injective and surjective and homeomorphism for 256 ECA rules, we make a general study for injective and surjective of all ECA rules by the two-sided mapδ. Finally, and show all homeomorphisms in 256 ECA rules.The rest of paper carefully analyzes the dynamical behaviors of rule 57. it is shown that rule 57 has two chaotic dynamical subsystems and presents very rich and complicat-ed dynamical properties. Especially, it is topologically mixing and possess the positive topological entropies on the two subsystems, Meanwhile, we estimate the infimum of topological entropy of f57 in Sz, by the subshift of finite type, it is concluded that rule 57 is chaotic in the sense of Li-Yorke and Devaney, aperiodic and irreducible. in addition, the discussion of periodic point and fixed point of rule 57 are studied.Finally, we make a brief summary on this work and prospect for further research.