Some Dynamical Properties Of Substitution Systems

The topology dynamical system originates from the research of Birkhoff.It developed greatly in 20 century .It is well know that the main task to topology dynamical system is clear the proximility of the point and the structure of the orbit of the point.There are several base conceptions which depict the complexity and disorder of the topology dynamical system,transitivity,rninimality,mixing,Li-Yorke chaos and distribution chaos,and so on.In 1975,Li and Yorke raised the conception "chaos" with exact mathematical language in their famous paper "Period three implies chaos" for the first time,which can be generalized as follow.Definition 1 Let X be a compact metric space, f:X→X a continuous map.The dynamical system (X,f) is called chaotic (in the sense of Li and Yorke ),if there exist an uncountable set S X,such that for all x, y∈S, x≠y,lim sup d(fⁿ(x),fⁿ(y))>0.we call it Li-Yorke chaos. In 1994, Schweizer and Smital defined another chaos by distribution function of distance between trajectories of two points.Definition 2 Let (X, d) be a compact metric space, f : X→X be continuous, if there exists a set D (?) X, such that, for x,y∈D with x≠y,(1) for someε> 0, F_x,y(ε) = 0;(2) for (?)_ε> 0, F_x,y^*(ε) = 1.then f is called to be distributionally chaotic on X. x,y is called to be a pair of points displaying distribution chaos.Li-Yorke chaos and distribution chaos are used to reflect the complexity and disorder of the topology dynamical system as well.The study of sequences arising from substitutions started in 1906 and lasted for nearly a century.Today they can be found in many branches of Math-matics,Computer Science and Physics.Particularly,there are many applications in such field as Dynamical System.Number Theory,Fractal Geometry,Complex Analysis,Harmonic Analysis etc.It plays an important role in those fields.In the province of Dynamical System,the dynamical systems arising from substitutions have been studied in detail.Definition 3 Let u be a sequence arising from substitutionη.In sequencespace∑, define X = orb(u,σ)(wheveσis the shift of∑).Then a dynamical system (X,σ) arises as a compact sub-systems of the symbolic system (∑,σ).we call it substitution system.In this paper ,we will introduce the some important results of substitution system,such as transitivity,miniality,topology mixing ,Li-Yorke chaos and distribution chaos.Theorem 1 Let substitutionηsatisfies the Hypothesis (see it in paper),then the fixed piont u =η^∞(0)∈R(σ) iff o occurs in u at least two times.Theorem 2 Let substitutionηsatisfies the Hypothesis,then the fixedpiont u =η^∞(0)∈R(σ) is a almost periodic point ofσiff orb(u,σ) is minimal iff for every i∈S,exist integer K(?)o,such thatη^K(i) contains the symbol o .Theorem 3 Letηbe primitive ,the second eigenvalueθ₂ satisfies |θ₂| >1,then orb(u,σ) is strong mixing iff GCD{ |ηⁿ(i) | | i,j∈S} = 1, (?)n≥1.Theorem 4 Letηbe primitive ,the second eigenvalueθ₂ satisfies |θ₂|<1,then orb(u,σ) is not weak mixing ,so does strong mixing.Finally,there are some important results about the minimal substitution with constant length on two letters.Theorem 5 Letηbe the minimal uncommon substitution with constant length n. Ifηsatisfies the hypothesis (C)(see it in paper),then the deduced map T have Li-Yorke chaotic set.Contrary,ifηdoes not satisfy the hypothesis (C),then T does not have Li-Yorke chaotic set.Theorem 5 Every substitution system is not (DC1) and (DC2).