【摘要】：近年來,許多學(xué)者研究了dendrite上的動(dòng)力系統(tǒng)性質(zhì),例如等度連續(xù)性、分離指數(shù)、軌道的收斂性、dendrite映射的極小集和中心深度等,但是dendrite上的等度連續(xù)性和中心深度尚有研究的空間.因此本文主要針對(duì)這兩個(gè)方面對(duì)dendrite展開研究.一個(gè)連續(xù)統(tǒng)是一個(gè)非空緊致連通的度量空間,而一個(gè)局部連通的不包含閉合曲線的連續(xù)統(tǒng)叫做dendrite.本文主要有以下結(jié)果.在第三章中,設(shè)D是一個(gè)有有限個(gè)分支點(diǎn)的dendrite以及f是D到它自身的連續(xù)映射.用R(f)和P(f)來分別表示回歸點(diǎn)集和非游蕩點(diǎn)集.設(shè)Ωo(f)=D,Ωn(f)=Ω(f|Ω-1(f))(對(duì)任意的n∈N).滿足Ωm(f)= Ωm+1(f)的最小的m ∈N ∪{∞}稱為f的深度.在本文中,證明了Ω3(f)=R(f)以及f的深度不超過3.而且找到了一個(gè)dendrite T,使其有兩個(gè)分支點(diǎn),以及找到f是T到它自身的連續(xù)映射,使得Ω3(f)=R(f)≠Ω2(f).在第四章中,設(shè)T是一個(gè)有有限個(gè)分支點(diǎn)的dendrite以及f是T到它自身的連續(xù)映射.用ω(χ,f)表示在f作用下的ω-極限集.非游蕩點(diǎn)集為Ω(χ,f)=.{存在點(diǎn)列{xk}(?)T以及遞增序列{nk}(?) N,使得且對(duì)dendrite上的等度連續(xù)性進(jìn)行研究,可得到如下等價(jià)結(jié)論：(1)f是等度連續(xù)的；(2)對(duì)任意的x∈T,ω(x,f)=Ω(x,f)；(3)對(duì)任意的x∈T,Ω(x,f)都是一條周期軌；(4)∩∞n=1fn(T)=P(f),并且對(duì)任意的x∈T,都有Card(ω(x,f))∞以及函數(shù)h:x →ω(x,f)(x∈f)連續(xù).另外,還構(gòu)造并證明了其他兩個(gè)特殊的dendrite:(1)存在一個(gè)dendrite D和f是D到D的連續(xù)映射,使得P(f)=Ω(f)≠D=CR(f),并且對(duì)任意的n ∈N, fn無湍流.(2)存在一個(gè)dendrite D和f是D到D的連續(xù)映射,滿足對(duì)某個(gè)的x∈D,存在y∈D,使得當(dāng)x∈Sα(y,f)時(shí),x (?) Sα(x,f).
[Abstract]:In recent years, many scholars have studied the properties of dynamical systems on dendrite, such as equal continuity, separation index, convergence of orbits, minimal sets of dendrite maps and central depth, etc. However, there is still room for research on equal continuity and central depth on dendrite. Therefore, this paper mainly focuses on these two aspects of the dendrite research. A continuum is a nonempty compact connected metric space, and a locally connected continuum without closed curves is called dendrite. The main results of this paper are as follows. In chapter 3, let D be a dendrite with finite fulcrum and f be a continuous mapping from D to itself. R (f) and P (f) are used to represent the regression point set and the non-wandering point set respectively. Let 惟 o (f) = D, 惟 n (f) = 惟 (f | 惟-1 (f) (). The minimum m 鈭，

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