Substitution Systems And Chaos

Topological dynamical systems is the main progressive state, and track topology. Transfer, trivial, Li-Yorke chaos and distribution chaos topological dynamicalsystems describes the complexity and degree of confusion.In 1975,Li and Yorke is the first attempt to use the strict definition of a chaotic mathematical language concepts:Definition 1. Let f is the metric space (X, d) to its continuous mapping, x, y inX. If satisfy:Says x, y for the Li-Yorke chaos points; If a subset of not less than two of any two different points is the Li-Yorke chaos point right, called for the pool the Li-Yorke chaos point set; If f has an uncountable Li-Yorke chaotic set, call Li-Yorke chaos.In1994,Schweizer and Smital have been corresponding with two track-the distance between the Schweizer-Smital function which definited distribution chaos:Definition 2. Let f is the metric space (X, d) to its continuous mapping, x, y inX. any t > 0 and any positive integer n, orderξ_n(f, x, y, t) = #{i | d(f~i(x), f~i(y)) 0, F(f, x, y,ε) = 0,(2)for (?)t>0, F~*(f,x,y,t) = 1.If a subset of not less than two of any two different points is the chaotic distribution point right, says the pool the distribution chaotic point set; If f has an uncountable distribution chaotic set, call distribution Chaos.We consider the following definition of the substitution system:Definition 3.ζhas a fixed point u in∑_s,for (?)k≥1,ζ~k(0) is the beginning of the symbol which belongs to u.Make X = orb(u,σ), T-σ|x, then the system (X, T) is the sub-system of the symbolic system (∑_s,σ),we call it the system which is deduced byζ,for short we call it substitution system.Substitution on the sequence of study has been carried out for a long time. In recent years, substitution system more in-depth way into the mathematics, Computers and the various branches of physics, particularly in the power system, number theory, fractal geometry, harmonic analysis, complex analysis, Portfolio theory, the form of the language in the field of play an increasingly greater role.This paper introduces some substitution system dynamic nature and explore its possible chaotic behavior, the results are as follows:(1) Substitution system introduced in the conditions (C) concept, then the substitutionζis primitive and isometric in the symbol space with two symbols,we prove:if theζsatisfy the condition (C),its induced mapping T have a Li-Yorke chaotic set with two points;Contrary,ifζdoes not satisfy the condition (C),then T does not have Li-Yorke chaotic set.(2) Forζnon-primitive and isometric in the symbol space with two symbols, if theζsatisfy the condition (C),its induced mapping T have a populated c Li-Yorke chaotic set,in particular,T is the Li-Yorke chaos;Contrary,ifζdoes not satisfy the condition (C),then T does not have Li-Yorke chaotic set.(3) The shift which is induced by any primitive and isometric substitution in the symbol space with two symbols does not exist distribution chaotic set,in particular,it is not the distribution chaos.