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数学物理学报, 2020, 40(4): 1061-1071 doi:

论文

关于Syndetic敏感和Multi敏感在Frechet空间的一个注记

姚权权,, 朱培勇

A Note About Syndetic Sensitivity and Multi-Sensitivity on Frechet Space

Yao Quanquan,, Zhu Peiyong

通讯作者: 姚权权, E-mail: yqqmath@163.com

收稿日期: 2019-03-12  

基金资助: 国家自然科学基金.  11501391

Received: 2019-03-12  

Fund supported: the NSFC.  11501391

摘要

(X,T)为动力系统,其中X是可分的Frechet空间和TXX是算子,首先证明了以下命题等价:(1)(X,T)是敏感的;(2)(X,T)是multi敏感;(3)(X,T)是multi-thick敏感;(4)(X,T)是thick敏感.不仅如此,还证明了如下命题:(X,T)是syndetic敏感当且仅当(X,T)是thickly syndetic敏感.如果(X,T)是syndetic传递,那么(X,T)是syndetic敏感.如果(X,T)F-超循环,那么(X,T)是thickly syndetic敏感.如果(X,T)是syndetic传递,那么(X,T)是传递敏感.最后证明了迭代动力系统也有类似的结果.

关键词: Multi敏感 ; Thickly Syndetic敏感 ; 迭代动力系统

Abstract

Let (X,T) be a linear dynamical system, where X is a separable Frechet space and T:XX is a operator. first we prove the following assertions are equivalent:(1) (X,T) is sensitive; (2) (X,T) is multi-sensitive; (3) (X,T) is multi-thickly-sensitive; (4) (X,T) is thickly sensitive. Then we prove the following propositions:(X,T) is syndetically sensitive if and only if (X,T) is thickly syndetically sensitive. If (X,T) is syndetic transitive, then (X,T) is syndetically sensitive. If (X,T) is frequently hypercyclic, then (X,T) is thickly syndetically sensitive. If (X,T) is syndetically transitive, then (X,T) is transitively sensitive. Moreover, the iterated function system has the similar results.

Keywords: Multi-sensitive ; Thickly syndetically sensitive ; Iterated function system

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本文引用格式

姚权权, 朱培勇. 关于Syndetic敏感和Multi敏感在Frechet空间的一个注记. 数学物理学报[J], 2020, 40(4): 1061-1071 doi:

Yao Quanquan, Zhu Peiyong. A Note About Syndetic Sensitivity and Multi-Sensitivity on Frechet Space. Acta Mathematica Scientia[J], 2020, 40(4): 1061-1071 doi:

1 引言和基本定义

Akin和Kolyada[1]引入了Li-Yorke敏感的概念将Li-Yorke混沌和初值敏感联系起来,并且证明了弱混合动力系统是Li-Yorke敏感的, Moothathu[16]介绍了一系列通过自然数子集表示的敏感,得到了syndetic敏感严格强于敏感,有限余敏感严格强于syndetic敏感.在文献[13]中,作者中讨论了关于一些敏感性和传递性的乘积系统,在文献[11, 12]中,刘等讨论了紧空间上的敏感, syndetic敏感, thickly syndetic敏感, Li-Yorke敏感.黄文等[8]介绍了传递敏感的概念,并且得到了multi敏感,敏感紧和传递敏感关于极小系统是等价的.不仅如此,黄文等[6, 7]证明了multi敏感, thick敏感, thickly syndetic敏感关于M系统是等价的, multi敏感和thick敏感关于传递系统是等价的,李[10]等介绍了超空间上(M(X),TM)上的multi-thick敏感, multi-syndetic敏感, multi-thickly-syndetic敏感. Godefroy[4]证明了超循环的算子是敏感的,刘,崔,王等专家分别研究了Banach空间各种算子[22-24],迭代动力系统可能是Hutchinson[9]初次介绍和被Barnsley[2]进一步发展. Ghane[3]研究了迭代动力系统的敏感性,并且证明了如果迭代动力系统是强传递的非极小系统,那么此动力系统是敏感的,更多的关于敏感性,乘积系统,迭代动力系统, Frechet空间请参考文献[5, 14, 15, 17-21].但是,上面许多专家集中研究动力系统(X,T),其中X是紧空间, T:XX是连续满射.本文则集中研究动力系统(X,T),其中X是Frechet空间, T:XX是算子.本文主要得到了如下结果.

首先证明了以下命题等价: (1) (X,T)是敏感的; (2) (X,T)是multi敏感; (3) (X,T)是multi-thick敏感; (4) (X,T)是thick敏感.

不仅如此,还证明了如下命题(X,T)是syndetic敏感当且仅当(X,T)是thickly syndetic敏感.如果(X,T)是syndetic传递,那么(X,T)是syndetic敏感.如果(X,T)F -超循环,那么(X,T)是thickly syndetic敏感.如果(X,T)是syndetic传递,那么(X,T)是传递敏感.最后证明了迭代动力系统也有类似的结果.

本部分先回忆一些动力系统的概念, Z+, R, C表示非负整数,实数,复数.

定义1.1  (1)集合SZ+称为syndetic的是指它具有有界的间距,即存在M>0,使得对任意的nZ+满足S{n,n+1,,n+M};

(2)集合SZ+称为thick的是指它包含了任意长的整数段,即存在序列ni,使得Si=1{ni,ni+1,,ni+i};

(3)集合SZ+称为thickly syndetic的是指对任意nZ+,存在一个syndetic集Sn={sn1<sn2<}使得Sj=1{snj,snj+1,,snj+n}.

注1.1  分别记全体syndetic集, thick集, thickly syndetic集为Fs, Ft, Fts.

定义1.2  (1)设p是定义在K (R或者C)上的向量空间X非负实值函数,满足如下条件:

(ⅰ) p(x+y)p(x)+p(y), x,yX;

(ⅱ) p(λx)=|λ|p(x), xX,αK;则称pX上的一个半范数.

(2)半范数序列(pn)n称为分离的,若对于任意的n1使得pn(x)=0,则x=0.

(3)向量空间X称为Frechet空间,若在其上定义一组分离的递增的半范数(pn)n,并且在度量d(x,y)=n=112nmin, x, y \in X 是完备的.

注1.2  (1)度量 d 称为不变的,是指对任意的 x, y, z \in X , d(x, y) = d(x + z, y + z) .

(2)令 X , Y 为Frechet空间,连续的线性函数 T:X \to Y 称为算子.记全体这样的算子 L(X, Y) ,如果 Y = X ,则称 T X 上的算子,记 L(X) = L(X, X) .

(3)令 X , Y 为Frechet空间,分别在其上定义一组分离的递增的半范数 {({p_n})_n} {({q_n})_n} ,定义空间 X \oplus Y = \{ (x, y):x \in X, y \in Y\} ,并且在其上定义一组分离的递增的半范数 {r_n}(x, y) = {p_n}(x) + {q_n}(y) ,那么空间 X \oplus Y = \{ (x, y):x \in X, y \in Y\} 为Frechet空间[5].

注1.3  (1)令 S:X \to X T:Y \to Y 分别为定义在Frechet空间 X Y 上的算子,那么定义算子 S \oplus T S \oplus T:X \oplus Y \to X \oplus Y , (S \oplus T)(x, y) = (Sx, Ty) , x \in X, y \in Y .

(2) (X, T) 称为线性动力系统,若 X 是一个可分的Frechet空间, T:X \to X 是一个算子.或简称 T 或者 T:X \to X 是一个线性动力系统.从此刻起,如果没特别指出,所有的算子定义在一个可分的Frechet空间上.

(3)设 A \subset {{\Bbb Z}^ + } ,定义 \underline {dens} (A) = \mathop {\lim }\limits_{N \to \infty } \inf \frac{{card\{ 0 \le n \le N;n \in A\} }}{{N + 1}} A 的上密度.

(4)对于动力系统 (X, T) , x \in X ,以及 U , V \subset X ,定义 N(x.U) = \{ n \in {{\Bbb Z}^ + };{T^n}x \in U\} . A - A = \{ a - b \ge 0;a, b \in A\} , N(U, V) = \{ n \in {{\Bbb Z}^ + };U \cap {T^{ - n}}V \ne \emptyset \} = \{ n \in {{\Bbb Z}^ + };{T^n}U \cap V \ne \emptyset \} .

定义1.3  (1) (X, T) 称为 F -超循环的,若存在 x \in X 使得对于任意的非空开集 U \subset X , \underline {dens} \{ n \in {{\Bbb Z}^ + }:{T^n}x \in U\} > 0 .并且称 x T F -超循环向量,记 T 的所有 F -超循环向量组成的集合为 FHC(T) .

(2) (X, T) 称为超循环的,若存在 x \in X 使得 orb(x, T) = \{ x, Tx, {T^2}x, \cdots \} 稠于 X .并且称 x T 的超循环向量,记 T 的所有超循环向量组成的集合为 HC(T) .

(3) (X, T) 称为传递的,是指对任意的非空开集 U, V \subset X N(U, V) \ne \emptyset .

(4) (X, T) 称为syndetic传递的,是指对任意的非空开集 U, V \subset X N(U, V) 是syndetic集.

注1.4  (1) (X, T) 是超循环的当且仅当 (X, T) 称为传递,并且 HC(T) 是一个稠密的 {G_\delta } .

(2)设 (X, T) 为动力系统,其中 X 上的度量为 d ,令 {N_T}(U, \delta ) = \{ n \in {{\Bbb Z}^ + }; 存在 {x_1}, {x_2} \in U 使得 d({T^n}{x_1}, {T^n}{x_2}) > \delta \}, B(x, r) = \{ y \in X;d(x, y) < r\} .

定义1.4  设 (X, T) 为线性动力系统,其中 X 上的度量为 d ,

(1) (X, T) 称为敏感的,是指存在 \delta > 0 (敏感常数)使得对任意 x \in X r > 0 ,能找到 y \in B(x, r) n \in {{\Bbb Z}^+} 满足 d({T^n}x, {T^n}y) > \delta .

(2) (X, T) 称为syndetic (或者thick或者thickly syndetic)敏感的,是指存在 \delta>0 (敏感常数)使得对任意非空开集 U \subset X 满足 {N_T}(U, \delta ) 是syndetic (或者thick或者thickly syndetic)集.

(3) (X, T) 称为multi敏感的,是指存在 \delta >0 (敏感常数)使得对任意非空开集 {U_1}, {U_2}, \cdots , {U_k} \subset X 满足 \bigcap\limits_{i = 1}^k {{N_T}({U_i}, \delta )} \ne \emptyset .

(4) (X, T) 称为multi-syndetic (或者multi-thick或者multi-thickly syndetic)敏感的,是指存在 \delta > 0 (敏感常数)使得对任意非空开集 {U_1}, {U_2}, \cdots , {U_k} \subset X 满足 \bigcap\limits_{i = 1}^k {{N_T}({U_i}, \delta )} 是syndetic (或者thick或者thickly syndetic)集.

(5) (X, T) 称为传递敏感的,是指存在 \delta > 0 (敏感常数)使得对任意非空开集 U, V, W \subset X 满足 {N_T}(W, \delta ) \cap N(U, V) \ne \emptyset .

定义1.5  设 (X, d) 是Frechet空间和 f_{\lambda} , \lambda \in \Lambda (其中 \Lambda 是有限非空数集)是 X 上的算子,那么 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 称为迭代动力系统.

注1.5  记所有的无限序列 \{\lambda_{i}\}_{i\in {{\Bbb Z}}^{+}} ,其中 {\lambda _i} \in \Lambda 组成的集合为 \Lambda^{{{\Bbb Z}}^{+}} ,例若 \sigma \in {\Lambda ^{{{\Bbb Z}_ + }}} ,可记 \sigma = \{\lambda_{0}, \lambda_{1}, \cdots \} .对任意 k\in {{\Bbb Z}}^{+} ,令 {\cal F}^{k} = \{X; f_{\lambda_{k-1}}\circ \cdots\circ f_{\lambda_{0}}| \lambda_{0}, \cdots, \lambda_{k-1}\in \Lambda \} , f_\sigma ^n(x) = {f_{{\lambda _{n - 1}}}} \circ \cdots \circ {f_{{\lambda _0}}}(x), x \in X , f_\sigma ^{ - n}(U) = \{ x \in X;f_\sigma ^nx \in U\} .

注1.6  对任意 \delta > 0 和任意非空开集 U, V\subset X ,定义

N_{{\cal F}}(U, \delta) = \{n\in {{\Bbb Z}}^+; \sup\limits_{\sigma \in \Lambda^{{{\Bbb Z}}^{+}}}\mbox{diam}f_{\sigma}^{n}(U)>\delta \}

N_{{\cal F}}(U, V) = \{n\in {{\Bbb Z}}^+; f_{\sigma}^{n}(U)\cap V\neq\emptyset\ \mbox{存在}\ \sigma \in \Lambda^{{{\Bbb Z}}^{+}}\}.

一般在算子理论经常写作 f_\sigma ^nx = f_\sigma ^n(x) .

定义1.6  (1)迭代动力系统 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 称为传递的,是指对任意的非空开集 U, V \subset X 满足 {N_{{\cal F}}}(U, V) \ne \emptyset .

(2)迭代动力系统 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 称为syndetic传递的,是指对任意的非空开集 U, V \subset X 满足 {N_{\cal F}}(U, V) 是syndetic集.

定义1.7  设 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 为迭代动力系统,其中 X 上的度量为 d ,

(1) {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 称为敏感的,是指存在 \delta > 0 (敏感常数)使得对任意 x \in X r > 0 ,能找到 y \in B(x, r) \sigma \in \Lambda^{{{\Bbb Z}}^{+}} n\in {{\Bbb Z}}^{+} 满足 d(f_{\sigma}^{n}(x), f_{\sigma}^{n}(y))>\delta .

(2) {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 称为syndetic (或者thick或者thickly syndetic)敏感的,是指存在 \delta > 0 (敏感常数)使得对任意非空开集 U \subset X 满足 N_{{\cal F}}(U, \delta) 是syndetic (或者thick或者thickly syndetic)集.

(3) {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 称为multi敏感的,是指存在 \delta > 0 (敏感常数)使得对任意非空开集 {U_1}, {U_2}, \cdots , {U_k} \subset X 满足 \bigcap\limits_{1\leq i\leq k}N_{{\cal F}}(U_{i}, \delta)\neq\emptyset .

(4) {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 称为multi-syndetic (或者multi-thick或者multi-thickly syndetic)敏感的,是指存在 \delta > 0 (敏感常数)使得对任意非空开集 {U_1}, {U_2}, \cdots , {U_k} \subset X 满足 \bigcap\limits_{1\leq i\leq k}N_{{\cal F}}(U_{i}, \delta) 是syndetic (或者thick或者thickly syndetic)集.

(5) {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 称为传递敏感的,是指存在 \delta > 0 (敏感常数)使得对任意非空开集 U, V, W \subset X 满足 {N_{{\cal F}}}(U, \delta ) \cap {N_{{\cal F}}}(U, V) \ne \emptyset .

2 Frechet空间上的敏感

命题2.1   (X, T) 是敏感的当且仅当 (X, T) 是multi敏感的.

  (必要性) 设 (X, T) 是敏感的(敏感常数为 \delta > 0 ),令 d X 上的任意传递不变的度量,对任意的 {r_1}, {r_2}, \cdots {r_k} > 0 {x_1}, {x_2}, \cdots, {x_k} \in X ,则 B({x_1}, {r_1}), B({x_2}, {r_2}), \cdots , B({x_k}, {r_k}) X 上的非空开集,因为 d 是传递不变的,所以

B({x_1}, {r_1}) = {x_1} + B(0, {r_1}), B({x_2}, {r_2}) = {x_2} + B(0, {r_2}), \cdots , B({x_k}, {r_k}) = {x_k} + B(0, {r_k}),

又因存在 B(0, l) \subset \bigcap\limits_{i = 1}^k {B(0, {r_i})} ,所以

\begin{equation} \emptyset \ne {N_T}(B(0, l), \delta ) \subset {N_T}(\bigcap\limits_{i = 0}^k {B(0, {r_i}), \delta } ), \end{equation}
(2.1)

对任意 n \in {N_T}(B(0, l), \delta ) ,则存在 {z_1}, {z_2} \in B(0, l) 满足 d({T^n}{z_1}, {T^n}{z_2}) > \delta ,易知

{x_i} + {z_1}, {x_i} + {z_2} \in B({x_i}, {r_i}), {\quad} i = 1, \cdots , k,

\begin{equation} d({T^n}({x_i} + {z_1}), {T^n}({x_i} + {z_2})) = d({T^n}{x_i} + {T^n}{z_1}, {T^n}{x_i} + {T^n}{z_2}) = d({T^n}{x_1}, {T^n}{x_2}) > \delta. \end{equation}
(2.2)

因此 n \in \bigcap\limits_{i = 1}^k {{N_T}(B({x_i}, {r_i}), \delta )} ,所以 (X, T) 是multi敏感的(敏感常数为 \delta > 0 ).

(充分性) 显然的.

命题2.2   (X, T) 是multi敏感的当且仅当 (X, T) 是multi-thick敏感的.

  (必要性) 方法(1)设 (X, T) 是multi敏感的(敏感常数为 \delta > 0 ),令 d X 上的任意传递不变的度量,对任意的 {r_1}, {r_2}, \cdots {r_k} > 0 {x_1}, {x_2}, \cdots {x_k} \in X ,则 B({x_1}, {r_1}), B({x_2}, {r_2}), \cdots , B({x_k}, {r_k}) X 上的非空开集,剩下的证明类似命题3.2[6].实际上因 (X, T) 是multi敏感的(敏感常数为 \delta > 0 ),对任意的 N \in {{\Bbb Z}^ + } ,则

\begin{equation} \bigcap\limits_{j = 0}^N {\bigcap\limits_{i = 1}^k {{N_T}({T^{ - j}}(B({x_i}, {r_i})), \delta )} } \ne \emptyset, \end{equation}
(2.3)

因此取 {n_N} \in \bigcap\limits_{j = 0}^N {\bigcap\limits_{i = 1}^k {{N_T}({T^{ - j}}(B({x_i}, {r_i})), \delta )} } {n_N} > N ,明显 \{ {n_N}, {n_N} - 1, \cdots , {n_N} - N\} \subset \bigcap\limits_{i = 1}^k {{N_T}(B({x_i}, {r_i}), \delta )} ,因此 (X, T) 是multi-thick敏感的(敏感常数为 \delta > 0 ).

方法(2)设 (X, T) 是multi敏感的(敏感常数为 \delta > 0 ),令 d X 上的任意传递不变的度量,对任意的 {r_1}, {r_2}, \cdots {r_k} > 0 {x_1}, {x_2}, \cdots {x_k} \in X ,则 B({x_1}, {r_1}), B({x_2}, {r_2}), \cdots , B({x_k}, {r_k}) X 上的非空开集,又因存在 B(0, r) \subset \bigcap\limits_{i = 0}^k {B(0, {r_i})} ,对任意的 N \in {{\Bbb Z}^ + } , {T^{ - i}}(B(0, r)) X 中的非空开集, 0 \le i \le N .因为 (X, T) 是multi敏感的(敏感常数为 \delta > 0 ),则 \bigcap\limits_{i = 0}^N {{N_T}({T^{ - i}}(B(0, r)), \delta )} \ne \emptyset ,因此取 {n_N} \in \bigcap\limits_{i = 0}^N {{N_T}({T^{ - i}}(B(0, r)), \delta )} {n_N} > N ,易知 \{ {n_N}, {n_N} - 1, \cdots , {n_N} - N\} \subset {N_T}(B(0, r), \delta ) ,因

\begin{equation} \{ {n_N}, {n_N} - 1, \cdots , {n_N} - N\} \subset {N_T}(B(0, r), \delta ) \subset \bigcap\limits_{i = 1}^k {{N_T}(B(0, {r_i}), \delta )}, \end{equation}
(2.4)

对任意的 n \in \bigcap\limits_{i = 1}^k {{N_T}(B(0, {r_i}), \delta )} ,则存在 z_1^i, z_2^i \in B(0, {r_i}) 满足 d({T^n}z_1^i, {T^n}z_2^i) > \delta ,又 {x_i} + z_1^i, {x_i} + z_2^i \in B({x_i}, {r_i})

\begin{equation} d({T^n}({x_i} + z_1^i), {T^n}({x_i} + z_2^i)) = d({T^n}{x_i} + {T^n}z_1^i, {T^n}{x_i} + {T^n}z_2^i) = d({T^n}z_1^i, {T^n}z_2^i) > \delta, \end{equation}
(2.5)

所以 n \in \bigcap\limits_{i = 1}^k {{N_T}} (B({x_i}, {r_i}), \delta ) ,因此 \{ {n_N}, {n_N} - 1, \cdots , {n_N} - N\} \subset \bigcap\limits_{i = 1}^k {{N_T}(B({x_i}, {r_i}), \delta )} ,故 (X, T) 是multi-thick敏感的(敏感常数为 \delta > 0 ).

(充分性) 显然的.

定理2.1  设 (X, T) 为线性动力系统,则以下命题等价:

(1) (X, T) 是敏感的;

(2) (X, T) 是multi敏感的;

(3) (X, T) 是multi-thick敏感的;

(4) (X, T) 是thick敏感的.

   (1) \Rightarrow (2) 见命题2.1; (2) \Rightarrow (3) 见命题2.2; (3) \Rightarrow (4) , (4) \Rightarrow (1) 显然的.

推论2.1   (X, T) 是敏感的当且仅当 (X, {T^n}) 是敏感的, n \in {{\Bbb Z}^ + } .

  (必要性) 由定理2.1知,若 (X, T) 是敏感的,则 (X, T) 是thick敏感,因此存在 \delta > 0 使得对任意非空开集 U \subset X , {N_T}(U, \delta ) 是thick集,所以对任意 n \in {{\Bbb Z}^ + } , \{ n, 2n, \cdots , ln, \cdots \} \cap {N_T}(U, \delta ) \ne \emptyset ,故 (X, {T^n}) 是敏感的.

(充分性) 显然的.

命题2.3   (X \oplus Y, T \oplus S) 是敏感的当且仅当 (X, T) 或者 (Y, S) 是敏感,其中 X 上的度量为 {d_X} Y 上的度量为 {d_Y} , X \oplus Y 上的度量为 {d_{X\oplus Y}}

  (必要性) 只需注意到

{d_X}({x_1}, {x_2}) + {d_Y}({y_1}, {y_2}) \ge \frac{1}{2}{d_{X \oplus Y}}(({x_1}, {x_2}), ({y_1}, {y_2})), \ ({x_1}, {y_1}), ({x_2}, {y_2}) \in X \oplus Y.

(充分性) 显然的.

定理2.2   (X, T) 是syndetic敏感当且仅当 (X, T) 是thickly syndetic敏感.

  (必要性) 设 (X, T) 是syndetic敏感(敏感常数为 \delta > 0 ),令 d X 上的任意传递不变的度量,对任意的 {x_0} \in X {r_0} > 0 ,则 B({x_0}, {r_0}) X 上的非空开集,且 B({x_0}, {r_0}) = {x_0} + B(0, {r_0}) ,对任意的 N \in {{\Bbb Z}^ + } , {T^{ - i}}(B({x_0}, {r_0})) X 上的非空开集,则存在 {x_i} \in X, {r_i} > 0 使得

\begin{equation} B({x_i}, {r_i}) = {x_i} + B(0, {r_i}) \subset {T^{ - i}}(B({x_0}, {r_0})), \end{equation}
(2.6)

i = 1, 2, \cdots , N ,又存在 B(0, l) \in \bigcap\limits_{i = 0}^N {B(0, {r_i})} ,因 {N_T}(B(0, l), \delta ) 是syndetic集,则对任意的 k \in {N_T}(B(0, l), \delta ) ,存在 {y_1}, {y_2} \in B(0, l) ,使得 d({T^k}{y_1}, {T^k}{y_2}) > \delta ,又 {x_i} + {y_1}, {x_i} + {y_2} \in B({x_i}, {r_i}) ,和

\begin{equation} d({T^k}({x_i} + {y_1}), {T^k}({x_i} + {y_2})) = d({T^k}{x_i} + {T^k}{y_1}, {T^k}{x_i} + {T^k}{y_2}) = d({T^k}{y_1}, {T^k}{y_2}) > \delta, \end{equation}
(2.7)

所以 k \in {N_T}(B({x_i}, {r_i}), \delta ) , i = 0, 1, \cdots , N ,因此 k \in \bigcap\limits_{i = 0}^N {{N_T}({T^{ - i}}B({x_0}, {r_0}), \delta )},

{n_N} \in \bigcap\limits_{i = 0}^N {{N_T}({T^{ - i}}B({x_0}, {r_0}), \delta )} \ \mbox{和}\ {n_N} > N,

明显 \{ {n_N}, {n_N} - 1, \cdots , {n_N} - N\} \subset {N_T}(B({x_0}, {r_0}), \delta ) ,令

A = \{ m \in {{\Bbb Z}^ + };m \in {N_T}(B(0, l), \delta ), m > N\} = \{ s_1^N < s_2^N < \cdots \},

易知 A 是syndetic集,又 \bigcup\limits_{j = 1}^\infty {\{ s_j^N, s_j^N - 1, \cdots , s_j^N - N\} } \subset {N_T}(B({x_0}, {r_0}), \delta ) ,故 (X, T) 是thickly syndetic敏感(敏感常数为 \delta > 0 ).

(充分性) 显然的.

命题2.4  若 (X, T) 是syndetic传递,那么 (X, T) 是syndetic敏感.

  令 a \in X\backslash \{ 0\} d X 上的任意传递不变的度量, \delta = \frac{1}{4}d(a, 0) > 0 ,令 U = B(a, \delta ) ,对任意的 \varepsilon > 0 x \in X ,则 B(x, \varepsilon ) = x + B(0, \varepsilon ) N(B(0, \varepsilon ), U) 是syndetic集,间距为 {M_1} ,下证 {N_T}(B(0, \varepsilon ), \delta ) 是syndetic集且间距为 {M_1} ,令 n \in {{\Bbb Z}^ + } ,取 j \in \{ 1, 2, \cdots , {M_1}\} v \in B(0, \varepsilon ) 满足 {T^{n + j}}v \in U ,则

\begin{equation} d({T^{n + j}}v, {T^{n + j}}0) = d({T^{n + j}}v, 0) > 2\delta > \delta. \end{equation}
(2.8)

n \in {{\Bbb Z}^ + } 是任意的和 j \le {M_1} ,故 {N_T}(B(0, \varepsilon ), \delta ) 是syndetic集,则 {N_T}(B(x, \varepsilon ), \delta ) 是syndetic集,实际上对任意 n \in {N_T}(B(0, \varepsilon ), \delta ) ,则存在 {a_1}, {a_2} \in B(0, \varepsilon ) 满足 d({T^n}{a_1}, {T^n}{a_2}) > \delta ,明显, x + {a_1}, x + {a_2} \in B(x, \varepsilon )

d({T^n}(x + {a_1}), {T^n}(x + {a_2})) = d({T^n}x + {T^n}{a_1}, {T^n}x + {T^n}{a_2}) = d({T^n}{a_1}, {T^n}{a_2}) > \delta,

n \in {N_T}(B(x, \varepsilon ), \delta ) ,因此 {N_T}(B(x, \varepsilon ), \delta ) 是syndetic集,故 (X, T) 是syndetic敏感.

推论2.2  若 (X, T) 是syndetic传递,那么 (X, T) 是传递敏感.

  因为 (X, T) 是syndetic传递,由定理2.2和命题2.4知 (X, T) 是thickly syndetic敏感.明显 (X, T) 是传递敏感.

推论2.3  若 (X, T) F -超循环的,那么 (X, T) 是thickly syndetic敏感.

  若 (X, T) F -超循环的,当然 (X, T) 是传递的,设 U , V X 的任意非空开集,则存在 {n_0} \ge 0 使得 {T^{{n_0}}}U \cap V \ne \emptyset ,因为 T 是连续的,则存在非空开集 {U_0} \subset U 满足 {T^{{n_0}}}{U_0} \subset V ,对任意的 x \in FHC(T) ,则 N(x, {U_0}) 上密度是正的,对任意 m, n \in N(x, {U_0}) , m \ge n , {T^{{n_0} + m - n}}({T^n}x) = {T^{{n_0}}}({T^m}x) \in V ,因此

{n_0} + (N(x, {U_0}) - N(x, {U_0})) \subset N({U_0}, V) \subset N(U, V),

由文献[18]知 N(x, {U_0}) - N(x, {U_0}) 是syndetic集,则 N(U, V) 是syndetic集.由定理2.2和命题2.4知因此 (X, T) 是thickly syndetic敏感.

3 迭代动力系统上的敏感

命题3.1  迭代动力系统 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是敏感的当且仅当 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是multi敏感的.

  (必要性) 设 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是敏感的(敏感常数为 \delta > 0 ),令 d X 上的任意传递不变的度量,对任意的 {r_1}, {r_2}, \cdots {r_k} > 0 {x_1}, {x_2}, \cdots, {x_k} \in X ,则 B({x_1}, {r_1}), B({x_2}, {r_2}), \cdots , B({x_k}, {r_k}) X 上的非空开集,因为 d 是传递不变的,所以

B({x_1}, {r_1}) = {x_1} + B(0, {r_1}), B({x_2}, {r_2}) = {x_2} + B(0, {r_2}), \cdots , B({x_k}, {r_k}) = {x_k} + B(0, {r_k}),

又因存在 B(0, l) \subset \bigcap\limits_{i = 1}^k {B(0, {r_i})} ,所以 \emptyset \ne N_{{\cal F}}(B(0, l), \delta ) ,和 N_{{\cal F}}(B(0, l), \delta ) \subset N_{{\cal F}}(\bigcap\limits_{i = 1}^k {B(0, {r_i})}, \delta ) ,对任意的 n \in N_{{\cal F}}(B(0, l), \delta ) ,则存在 {z_1}, {z_2} \in B(0, l) , \sigma \in \Lambda^{{{\Bbb Z}}^{+}} 满足 d({T_\sigma ^n}{z_1}, {T_\sigma ^n}{z_2}) > \delta ,易知 {x_i} + {z_1}, {x_i} + {z_2} \in B({x_i}, {r_i}), i = 1, \cdots , k ,

\begin{equation} d({T_\sigma ^n}({x_i} + {z_1}), {T_\sigma ^n}({x_i} + {z_2})) = d({T_\sigma ^n}{x_i} + {T_\sigma ^n}{z_1}, {T_\sigma ^n}{x_i} + {T_\sigma ^n}{z_2}) = d({T_\sigma ^n}{x_1}, {T_\sigma ^n}{x_2}) > \delta, \end{equation}
(3.1)

因此 n \in \bigcap\limits_{i = 1}^k {{N_{{\cal F}}}(B({x_i}, {r_i}), \delta )} ,所以 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是multi敏感的(敏感常数为 \delta > 0 ).

(充分性) 显然的.

命题3.2  迭代动力系统 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是multi敏感的当且仅当 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是multi-thick敏感的.

  (必要性) 设 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是multi敏感的(敏感常数为 \delta > 0 ),令 d X 上的任意传递不变的度量,对任意的 {r_1}, {r_2}, \cdots {r_k} > 0 {x_1}, {x_2}, \cdots {x_k} \in X ,则 B({x_1}, {r_1}), B({x_2}, {r_2}), \cdots , B({x_k}, {r_k}) X 上的非空开集,又因存在 B(0, r) \subset \bigcap\limits_{i = 1}^k {B(0, {r_i})} ,对任意的 N \in {{\Bbb Z}^ + } ,则

\begin{eqnarray*} &&0 \in (B(0, r) \cap (\bigcap\limits_{{i_1} \in \Lambda } {{f_{{i_1}}}^{ - 1}(B(0, r))} ) \cap (\bigcap\limits_{{i_1}, {i_2} \in \Lambda } {{{({f_{{i_1}}} \circ {f_{{i_2}}})}^{ - 1}}(B(0, r))} ) \cap \cdots \\ && \cap (\bigcap\limits_{{i_{1, }}{i_2}, \cdots , {i_N} \in \Lambda } {{{({f_{{i_1}}} \circ {f_{{i_2}}} \circ \cdots \circ {f_{{i_N}}})}^{ - 1}}(B(0, r))} )) \ne \emptyset, \end{eqnarray*}

则存在

\begin{eqnarray*} &&B(0, l) \subset (B(0, r) \cap (\bigcap\limits_{{i_1} \in \Lambda } {{f_{{i_1}}}^{ - 1}(B(0, r))} ) \cap (\bigcap\limits_{{i_1}, {i_2} \in \Lambda } {{{({f_{{i_1}}} \circ {f_{{i_2}}})}^{ - 1}}(B(0, r))} ) \cap \cdots \\ && \cap (\bigcap\limits_{{i_{1, }}{i_2}, \cdots , {i_N} \in \Lambda } {{{({f_{{i_1}}} \circ {f_{{i_2}}} \circ \cdots \circ {f_{{i_N}}})}^{ - 1}}(B(0, r))} )), \end{eqnarray*}

{N_{{\cal F}}}(B(0, l), \delta ) \ne \emptyset ,取 {n_N} \in {N_{{\cal F}}}(B(0, l), \delta ) {n_N} > N ,则存在 {y_1}, {y_2} \in B(0, l) \sigma \in \Lambda^{{{\Bbb Z}}^{+}} 满足 d(T_\sigma ^{{n_N}}{y_1}, T_\sigma ^{{n_N}}{y_2}) > \delta ,同时

\begin{eqnarray*} &&B(0, l) \subset (B(0, r) \cap (\bigcap\limits_{{i_1} \in \Lambda } {{f_{{i_1}}}^{ - 1}(B(0, r))} ) \cap (\bigcap\limits_{{i_1}, {i_2} \in \Lambda } {{{({f_{{i_1}}} \circ {f_{{i_2}}})}^{ - 1}}(B(0, r))} ) \cap \cdots \\ && \cap (\bigcap\limits_{{i_{1, }}{i_2}, \cdots , {i_N} \in \Lambda } {{{({f_{{i_1}}} \circ {f_{{i_2}}} \circ \cdots \circ {f_{{i_N}}})}^{ - 1}}(B(0, r))} )), \end{eqnarray*}

易知 \{ {n_N}, {n_N} - 1, \cdots , {n_N} - N\} \subset {N_{{\cal F}}}(B(0, r), \delta ) ,又对任意 n \in {N_{{\cal F}}}(B(0, r), \delta ) ,存在 {z_1}, {z_2} \in B(0, r) {\sigma _1}\in \Lambda^{{{\Bbb Z}}^{+}} 满足 d(T_{{\sigma _1}}^n{z_1}, T_{{\sigma _1}}^n{z_2}) > \delta ,因此 {x_i} + {z_1}, {x_i} + {z_2} \in B({x_i}, {r_i})

\begin{equation} d(T_{{\sigma _1}}^n({x_i} + {z_1}), T_{{\sigma _1}}^n({x_i} + {z_2})) = d(T_{{\sigma _1}}^n{x_i} + T_{{\sigma _1}}^n{z_1}, T_{{\sigma _1}}^n{x_i} + T_{{\sigma _1}}^n{z_2}) = d(T_{{\sigma _1}}^n{z_1}, T_{{\sigma _1}}^n{z_2}) > \delta, \end{equation}
(3.2)

所以 n \in {N_{{\cal F}}}(B({x_i}, {r_i}), \delta ) , i = 1, 2, \cdots , k ,故

\begin{equation} \{ {n_N}, {n_N} - 1, \cdots , {n_N} - N\} \subset \bigcap\limits_{i = 1}^k {{N_{{\cal F}}}(B({x_i}, {r_i}), \delta )}, \end{equation}
(3.3)

因此 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是multi-thick敏感的(敏感常数为 \delta > 0 ).

(充分性) 显然的.

定理3.1  设 {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 为迭代动力系统,则以下命题等价:

(1) {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是敏感的;

(2) {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是multi敏感的;

(3) {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是multi-thick敏感的;

(4) {\cal F} = \{X; f_{\lambda}|\lambda \in\Lambda \} 是thick敏感的.

   (1) \Rightarrow (2) 见命题3.1; (2) \Rightarrow (3) 见命题3.2; (3) \Rightarrow (4) , (4) \Rightarrow (1) 显然的.

推论3.1  迭代动力系统 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是敏感的当且仅当 {F_n} = \{ X;{f_{{\lambda _1}}} \circ {f_{{\lambda _2}}} \circ \cdots \circ {f_{{\lambda _n}}}|{\lambda _i} \in \Lambda, 1 \le i \le n\} 是敏感的, n \in {{\Bbb Z}^ + } .

  (必要性) 由定理3.1知,若 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是敏感的,则 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是thick敏感,因此存在 \delta > 0 使得对任意非空开集 U \subset X , {N_{{\cal F}}}(U, \delta ) 是thick集,所以对任意 n \in {{\Bbb Z}^ + } , {N_{{\cal F}}}(U, \delta ) \cap \{ n, 2n, \cdots , kn, \cdots \} \ne \emptyset ,故 {N_{{{\cal F}_n}}}(U, \delta ) \ne \emptyset ,故 {F_n} = \{ X;{f_{{\lambda _1}}} \circ {f_{{\lambda _2}}} \circ \cdots \circ {f_{{\lambda _n}}}|{\lambda _i} \in \Lambda, 1 \le i \le n\} 是敏感的.

(充分性) 显然的.

命题3.3  设 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} {\cal G} = \{Y; g_{\gamma}| \gamma \in \Lambda_1\} 是两个迭代动力系统分别定义在Frechet空间 X Y ,则 {\cal F}\times {\cal G} 是敏感的当且仅当 {\cal F} 或者 {\cal G} 是敏感,其中 X 上的度量为 {d_X} Y 上的度量为 {d_Y} , X \oplus Y 上的度量为 {d_{X\oplus Y}} .

  (必要性) 只需注意到

{d_X}({x_1}, {x_2}) + {d_Y}({y_1}, {y_2}) \ge \frac{1}{2}{d_{X \oplus Y}}(({x_1}, {x_2}), ({y_1}, {y_2})), \ ({x_1}, {y_1}), ({x_2}, {y_2}) \in X \oplus Y.

(充分性) 显然的.

定理3.2   {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是syndetic敏感当且仅当 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是thickly syndetic敏感.

  (必要性) 设 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是syndetic敏感(敏感常数为 \delta > 0 ),令 d X 上的任意传递不变的度量,对任意的 \varepsilon > 0 x \in X ,则 B(x, \varepsilon ) X 上的非空开集,且 B(x, \varepsilon ) = x + B(0, \varepsilon ) ,对任意的 N \in {{\Bbb Z}^ + } ,有

\begin{eqnarray*} &&0 \in B(0, \varepsilon ) \cap (\bigcap\limits_{{\lambda _1} \in \Lambda } {f_{{\lambda _1}}^{ - 1}} (B(0, \varepsilon ))) \cap {(\bigcap\limits_{{\lambda _i} \in \Lambda, 1 \le i \le 2} {({f_{{\lambda _1}}} \circ {f_{{\lambda _2}}}} )^{ - 1}}(B(0, \varepsilon ))) \cdots \\ && \cap (\bigcap\limits_{{\lambda _i} \in \Lambda, 1 \le i \le N} {{{({f_{{\lambda _1}}} \circ {f_{{\lambda _2}}} \circ \cdots \circ {f_{{\lambda _N}}})}^{ - 1}}(B(0, \varepsilon ))} ), \end{eqnarray*}

因此存在

\begin{eqnarray*} &&B(0, r) \subset B(0, \varepsilon ) \cap (\bigcap\limits_{{\lambda _1} \in \Lambda } {f_{{\lambda _1}}^{ - 1}} (B(0, \varepsilon ))) \cap {(\bigcap\limits_{{\lambda _i} \in \Lambda, 1 \le i \le 2} {({f_{{\lambda _1}}} \circ {f_{{\lambda _2}}}} )^{ - 1}}(B(0, \varepsilon ))) \cdots \\ && \cap (\bigcap\limits_{{\lambda _i} \in \Lambda, 1 \le i \le N} {{{({f_{{\lambda _1}}} \circ {f_{{\lambda _2}}} \circ \cdots \circ {f_{{\lambda _N}}})}^{ - 1}}(B(0, \varepsilon ))} ), \end{eqnarray*}

{n_N} \in {N_{{\cal F}}}(B(0, r), \delta ) {n_N} > N ,因为

\begin{eqnarray*} &&B(0, r) \subset B(0, \varepsilon ) \cap (\bigcap\limits_{{\lambda _1} \in \Lambda } {f_{{\lambda _1}}^{ - 1}} (B (0, \varepsilon ))) \cap {(\bigcap\limits_{{\lambda _i} \in \Lambda, 1 \le i \le 2} {({f_{{\lambda _1}}} \circ {f_{{\lambda _2}}}} )^{ - 1}}(B(0, \varepsilon ))) \cdots \\ && \cap (\bigcap\limits_{{\lambda _i} \in \Lambda, 1 \le i \le N} {{{({f_{{\lambda _1}}} \circ {f_{{\lambda _2}}} \circ \cdots \circ {f_{{\lambda _N}}})}^{ - 1}}(B(0, \varepsilon ))} ), \end{eqnarray*}

显然, \{ {n_N}, {n_N} - 1, \cdots , {n_N} - N\} \subset {N_{{\cal F}}}(B(0, \varepsilon ), \delta ) ,对任意的 n \in {N_{{\cal F}}}(B(0, \varepsilon ), \delta ) ,存在 {x_1}, {x_2} \in B(0, \varepsilon ) \sigma \in \Lambda^{{{\Bbb Z}}^{+}} 满足 d(f_\sigma ^n{x_1}, f_\sigma ^n{x_2}) > \delta ,则 x + {x_1}, x + {x_2} \in B(x, \varepsilon )

\begin{equation} d(f_\sigma ^n(x + {x_1}), f_\sigma ^n(x + {x_2})) = d(f_\sigma ^nx + f_\sigma ^n{x_1}, f_\sigma ^nx + f_\sigma ^n{x_2}) = d(f_\sigma ^n{x_1}, f_\sigma ^n{x_2}) > \delta, \end{equation}
(3.4)

n \in {N_{{\cal F}}}(B(x, \varepsilon ), \delta ) ,令

\begin{equation} A = \{ m \in {{\Bbb Z}^ + };m \in {N_{{\cal F}}}(B(0, r ), \delta ), m > N\} = \{ s_1^N < s_2^N < \cdots \}, \end{equation}
(3.5)

\bigcup\limits_{j = 1}^\infty {\{ s_j^N, s_j^N - 1, \cdots , s_j^N - N\} } \subset {N_{{\cal F}}}(B(x, \varepsilon ), \delta ) ,故 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是thickly syndetic敏感(敏感常数为 \delta > 0 ).

(充分性) 显然的.

引理3.1 若迭代动力系统 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是传递的,则集合 A \subset X 是一个稠密的 {G_\delta } -集,其中对任意 x \in A , \bigcup\limits_{\sigma \in {\Lambda ^{{{\Bbb Z}}^ {+} }}} {\bigcup\limits_{n = 0}^\infty {f_\sigma ^nx} } 稠于 X .

  因为 X 是可分的,所以 X 存在可数稠子集 \{ {y_j};j \ge 1\} ,且 B({y_j}, \frac{1}{m}) , m, j \ge 1 ,是 X 的一组可数基 {({U_k})_{k \ge 1}} ,因此, x \in A 当且仅当对任意 k \ge 1 ,存在 n \ge 0 \sigma \in {\Lambda ^{{{\Bbb Z}}^ {+ }}} 满足 f_\sigma ^nx \in {U_k} .也就是说 A = \bigcap\limits_{k = 1}^\infty {\bigcup\limits_{\sigma \in {\Lambda ^{{{\Bbb Z}^ + }}}} {\bigcup\limits_{n = 0}^\infty {f_\sigma ^{ - n}({U_k})} } } ,因 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是传递的,因此 \bigcup\limits_{\sigma \in {\Lambda ^{{{\Bbb Z}}^ {+ }}}} {\bigcup\limits_{n = 0}^\infty {f_\sigma ^{-n}({U_k})} } , k \ge 1 ,开稠于 X ,由Baire定理知 A 是一个稠密的 {G_\delta } -集.

命题3.4  若迭代动力系统 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是syndetic传递,那么 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是syndetic敏感.

  令 a \in X\backslash \{ 0\} d X 上的任意传递不变的度量, \delta = \frac{1}{4}d(a, 0) > 0 ,令 U = B(a, \delta ) ,对任意的 \varepsilon > 0 x \in X ,则 B(x, \varepsilon ) = x + B(0, \varepsilon ) {N_{{\cal F}}}(B(0, \varepsilon ), U) 是syndetic集,间距为 {M_1} ,下证 {N_{{\cal F}}}(B(0, \varepsilon ), \delta ) 是syndetic集且间距为 {M_1} ,令 n \in {{\Bbb Z}^ + } ,取 j \in \{ 1, 2, \cdots , {M_1}\} \sigma \in {\Lambda ^{{{\Bbb Z}_ + }}} , a \in B(0, \varepsilon ) 满足 f_\sigma ^{n + j}a \in U ,则

\begin{equation} d(f_\sigma ^{n + j}a, f_\sigma ^{n + j}0) = d(f_\sigma ^{n + j}a, 0) > 2\delta > \delta, \end{equation}
(3.6)

n \in {{\Bbb Z}^ + } 是任意的和 j \le {M_1} ,因此 {N_{\cal F}}(B(0, \varepsilon ), \delta ) 是syndetic集,则 {N_{{\cal F}}}(B(x, \varepsilon ), \delta ) 是syndetic集,实际上对任意 n \in {N_{{\cal F}}}(B(0, \varepsilon ), \delta ) ,则存在 {x_1}, {x_2} \in B(0, \varepsilon ) , \sigma \in {\Lambda ^{{{\Bbb Z}^ + }}} 满足 d(f_\sigma ^n{x_1}, f_\sigma ^n{x_2}) > \delta ,明显 x + {x_1}, x + {x_2} \in B(x, \varepsilon ) ,且

\begin{equation} d(f_\sigma ^n(x + {x_1}), f_\sigma ^n(x + {x_2})) = d(f_\sigma ^nx + f_\sigma ^n{x_1}, f_\sigma ^nx + f_\sigma ^n{x_2}) = d(f_\sigma ^n{x_1}, f_\sigma ^n{x_2}) > \delta, \end{equation}
(3.7)

所以 n \in {N_{{\cal F}}}(B(x, \varepsilon ), \delta ) ,因此 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是syndetic敏感.

推论3.2 (1)若 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是syndetic传递的,则 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是thickly syndetic敏感.

(2)若 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是syndetic传递的,则 {\cal F} = \{X; f_{\lambda}| \lambda \in \Lambda\} 是传递敏感.

  (1)由命题3.4和定理3.2, (2)由推论3.2(1)可得.

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