研究 (
F1,
F2)-攀援集的迭代不变性.定义了与正整数
k 相关的 Furstenberg 族的性质
P(
k) 和
Q(
k).指出: 对任意介于 0, 1 之间的实数
s 而言, Furstenberg 族
具有性质
P(
k) 和
Q(
k), 其中
表示非负整数集的所有上密度不小于
s的无限子集构成的集族. 据此证明了: 对任意正整数
k,
S 为系统(
X,
f) 的
-攀援集当且仅当
S为系统 (
X,
fk) 的
-攀援集,其中
s,t是介于 0,1 之间的任意给定的实数.
Abstract
This paper deals with the invariance (
F1,
F2)-scrambled set under iterations. For a positive integer
k, the properties
P(
k) and
Q(
k) of Furstenberg families are introduced. It is shown that for any
s∈[0,1], the Furstenberg family
has the properties
P(
k) and
Q(
k), where
denotes the family of all infinite subsets of
whose upper density is not less than
s. Furthermore, we prove that for any positive integer
k,
S is an
-scrambled set of (
X,f) if and only if
S is an
-scrambled set of (
X,
fk), where
s,t∈[0,1].
关键词
族 /
攀援集 /
混沌
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Key words
Furstenberg family /
scrambled set /
chaos
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参考文献
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[8] Blanchard F., Huang W., Snoha L., Topological size of scrambled sets, Colloq. Math., 2008, 110: 293--361.
[9] Wang L. D., Huan S. M., Huang G. F., A note on Schweizer-Smital chaos, Nonlinear Analysis, 2008, 68: 1682--1686.
[10] Yuan D. L., On Distributional Chaos, South China Normal University Ph.D.Thesis, in Chinese, 2008.
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脚注
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基金
国家自然科学基金资助项目(10771079);广州市属高校科技计划项目(08C016)
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