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1、广西师范大学 硕士学位论文 动力系统中的等度连续性及稠密集上的动力性质 姓名：徐香萍 申请学位级别：硕士 专业：应用数学 指导教师：赵俊玲 20090401 ? ?:?:? ?:?:?: 2006? ? ?.? ?: ?,? ?. ?,?: ?2.1.1?f?,?f?,?CR(f) = UA(f).?CR(f) = P(f)(?2.1.1). ?2.1.2?f?X? T n=1f n(X) = UA(f). ?2.1.3 f?X?,?W(f)? ?. ?2.1.4?f?X?,?f|T n=1f n(X) ?. ?f f?,?: ?2.2.1?X?, f : X X?, f : X X X X
2、f(x,y) = (f(x),f(y),(x X,y X),?(f f) = (f) (f), W(f f) = W(f) W(f), R(f f) = R(f) R(f), AP(f f) = AP(f) AP(f). ?2.2.2?X?f : X X?,?f?,?f f ?. ?fk?,?: ?2.3.1?X?, f : X X?,?k Z+, (f) = (fk),W(f) = W(fk). ?2.3.2?X?, f : X X?,?fk(k Z+) ?. ?. ?,?: ?3.1.1?(X,d)?, f : X X?, Y?X?, ?f|Y: Y X?f : X X?. ?3.1.2?
3、(X,d)?, f : X X?, Y?X?, f|Y: Y X?f : X X?. I ?3.1.3?(X,d)?, f : X X?, Y?X?, ?f|Y: Y X?,?f : X X?. ?,?: ?3.2.1?(X,d)?, f : X X?, Y?X?, ?f|Y: Y X?,?f : X X?. ?3.2.2?(X,d)?, f : X X?, Y?X?, ?f|Y: Y X?,?f : X X?. ?3.2.3?(X,d)?, f : X X?, Y?X?, ?f|Y: Y X?POTP,?f : X X?POTP. ?:?;?; ?;? II Equicontinuity i
4、n Dynamical Systems and Dynamical Properties in Dense Set Postgraduate: Xu XiangpingSupervisor: A.Prof. Zhao Junling Major: Applied Mathematics Research fi elds: Dynamical SystemsGrade: 2006 Abstract This dissertation did the research on the equicontinuity in dynamical systems and the dynam- ical pr
5、operties in the dense set. The paper is organized as follows: In chapter 1, we briefl y outline the development and applications of dynamical systems , then we related basic concepts in equicontinuous map. In chapter 2, we study the equicontinuous map on compact metric space X and prove some theorem
6、s. Theorem2.1.1 If f is equicontinuous, then its chain recurrent set of f is equal to the uniform almost periodic point set(i.e.CR(f) = UA(f). We give an examples (eg. 2.1.1) for this conclusion that can not be further strengthened to CR(f) = P(f) . Theorem2.1.2 If f is equicontinuous, then T n=1f n
7、(X) = UA(f). In addition, we give a necessary and suffi cient condition of equicontinuous maps. Theorem2.1.3 f is a continuous self-maps on compact metric space X , then f is equicon- tinuous on X if and only if each point of W(f) is equicontinuous points. Theorem2.1.4 If f is equicontinuous, then f
8、|T n=1f n(X)is a homeomorphism. In section 2, we study the product map f f. The main results as follows: Theorem2.2.1If f is equicontinuous map on compact metric space X . f : XX XX f(x,y) = (f(x),f(y),(x X,y X), then (f f) = (f) (f), W(f f) = W(f) W(f), R(f f) = R(f) R(f), AP(f f) = AP(f) AP(f). Th
9、eorem2.2.2 f is a continuous self-maps on compact metric space X . If f is equicontin- uous, then f f is not topological transitivity. In section 3, we study the fk. The main results as follows: Theorem2.3.1 X is compact metric space . If f is equicontinuous , then k Z+, (f) = (fk),W(f) = W(fk). The
10、orem2.3.2 X is connected compact metric space . If f is equicontinuous and topological transitivity , then fk(k Z+) is topological transitivity. In chapter 3, we study the dynamical properties in the dense set. The result as follows: III Theorem 3.1.1 Let (X,d) be a compact metric space. f : X X is
11、a continuous map, Y is a dense subset of X . Then f|Y: Y X is equicontinuous can not obtain f : X X is equicontinuous. Theorem 3.1.2 Let (X,d) be a compact metric space. f : X X is a continuous map, Y is a dense subset of X . Then f|Y: Y X is expansive map can not obtain f : X X is expansive map. Th
12、eorem 3.1.3 Let (X,d) be a compact metric space. f : X X is a continuous map, Y is a dense subset of X . If f|Y: Y X is sensitive dependence on initial conditions, then f : X X topological transitivity. Theorem 3.2.1 Let (X,d) be a compact metric space. f : X X is a continuous map, Y is a dense subs
13、et of X . If f|Y: Y X is topological transitivity, then f : X X is topological transitivity. Theorem 3.2.2 Let (X,d) be a compact metric space. f : X X is a continuous map, Y is a dense subset of X . If f|Y: Y X is minimal , then f : X X is minimal. Theorem 3.2.3 Let (X,d) be a compact metric space, f : X X is a continuous map, Y is a dense subset of X . If f|Y: Y X has POTP, then f : X X has POTP. Keywords: Equicontinuity; Uniform almost periodic point; -limit point; Dense set IV ? ?:? ?.?,? ?.?,? ?.?. ?:?: ? ?
