符号空间有限型子转移的稠密混沌

DENSE CHAOS FOR SUBSHIFTS OF FINITE TYPE IN SYMBOLIC SPACE

设T是紧致度量空间(X,ρ)上的连续自映射.说T是稠密混沌的(通有混沌的),如果{(x,y)∈X2|limsupρ(Tn(x),Tn(y))>0,liminfρ(Tn(x),Tn(y))=0}是X2的一个稠密集(相应地,剩n→∞n→∞余集).证明了:对于由不可约的0,1-方阵A决定的有限型子转移σA而言,σA是稠密混沌的,σA是通有混沌的以及σA是拓扑混合的,这些叙述均等价.并进一步指出,σA是稠密混沌的与σA是Li-Yorke混沌的等价当且仅当方阵A的阶数少于4.

Suppose that T is a continuous self map on a compact metric space endowed with a metric ρ. It is said that T is densely chaotic (generically chaotic),if {(x,y)∈X^2|(lim sup)n?→∞ (ρ(T()~n(x),)T()~n(y))>0,(lim inf)n→∞ ρ(T()~n(x),T()~n(y))=0} is dense (residual, resp.) in X^2. In the present paper, it is shown that for σ_A, the subshift of finite type determined by an irreducible square matrix A consisting of 0's and 1's, the following three statements are equivalent: σ_A is densely chaotic, σ_A is generically chaotic and σ_A is topologically mixing. Moreover, that σ_A is densely chaotic is equivalent to that σ_A is Li-Yorke chaotic if and only if the order of the square matrix A is less than 4.