符号动力系统的扩充系统的分布混沌性

Distributional Chaos for The Extended System of Symbolic Dynamical System

设(∑,σ)是两个符号的单边符号动力系统,(X,f)是紧致系统.如果存在连续满射h:X→∑,使得hof=σoh,则称(X,f)是(∑,σ)的扩充系统.本文研究(∑,σ)的扩充系统(X,f)的分布混沌性,通过在(∑,σ)中构造合适的符号序列,在扩充系统(X,f)中构造出了一个不可数的分布混沌集.证明了:若彐x∈∑,使得#h^(-1)(x)=1,则f是分布混沌的.

Let（Σ,σ） be the one-sided symbolic dynamical system which has two symbols A compact system（X,f） is called an extended system of（Σ,σ） if there exists a continuous map h of X onto the symbolic space E such that h o f = σ o h.In this paper,we study the distributionally chaotic behaviour of the extended system（X,f）.First,we construct special symbolic sequences in（Σ,σ）,then using these symbolic sequences and the topological semiconjugacy between（Σ,σ） and（X,f）,we construct a uncountable distributionally chaotic set in the extended system（X,f）.It is proved that if there is a point x of Σ which preimage of h is a single point,then the extended system（X,f）is distributionally chaotic.