Some Subshitfs On Symbolic Spaces

In the study of dynamical systems, the shift on symbolic spaces and itssubshift is of special importance. It is one of the most important research topicin dynamical systems.In this paper,(Σ, ρ) is the one-sided symbolic space with two symbols.σ is continuous and called the shift on Σ. If X Σ is closed and σ(X) X, wecall (X, σ|X) or σ|Xa subshift of σ.Firstly, we constructed the following subshift.Let A=a1a2anbe a tuple over S={0,1}. Define the inverse of A to beTake an arbitrary tuple A1. Let A2be an arrangement of A1andAˉ1, sayDefine inductively the tuples A2, A3, such that Anis anarrangement of all the tuples of the finite setLet a=A1A2, then ω(a, σ) is uncountably minimal set. Let T=ω(a, σ),then σ|Tis a minimal subshift.We proved that:(1)σ|Tis Wiggins chaotic,(2)σ|Tis Martelli chaotic,(3)σ|Tis distributionally chaotic, (4)σ|Thas zero topological entropy, i.e. σ|Tis not topologically chaotic.(5)σ|Tis topologically weakly mixing,(6) σ|Tis strictly ergodic.Furthermore, if there exists a conjugate which is from f to σ, there exists aminimal system f|Dsuch that(1)f|Dis Wiggins chaotic,(2)f|Dis Martelli chaotic,(3)f|Dis strictly ergodic.Secondly, we studied the other subshift on symbolic spaces—substitutionsubshift.Let S={0,1}, S is the set of all the tuples on S. Every map that is fromand it satisfies the following condition.Under the hypothesis (H), ζ has a fixed point beginning with0in Σ, denoted, then T: Xζ→Xζis a subsystemof substitution σ, is called the substitution subshift induced by ζ.If0η(1), we call η primitive, and non-primitive otherwise.When p=q, for ζ is both primitive and non-primitive, we proved the fol-lowing results.Let ζ satisfy condition (C).(C) there exists some distinct s1, s2, t∈{0,1,, n1}, such thatWhen ζ is non-primitive, we got the following result.If q≥p, there is no DC pair of T.Now there are many results about Devaney chaos, distributional chaos andtopological chaos. One easily gives an example which is either distributional ortopological chaos but not Devaney chaos. However, the inverse implications arenot such evident. To show that Devaney chaos does not imply distributionalchaos, Oprocha constructed a Devaney chaotic subshift without DC pairs, how-ever, he did not give a strict proof. where ni=2i(5|qi|+5i+5). Let u be the limit of the sequence {pi}∞i=1and Xthe closure of the orbit of u. Then (X, σ|X) is a subshift and it is Devaney chaos.Furthermore, we proved that(1)There doesn't exist DC pairs of σ|X,(2)ent(σ|X)=0, i.e. σ|Xis not topological chaos.It is a unified proof for the results of Weiss in1971and Oprocha.