On Limits Sets in Dynamical Systems
On Limits Sets in Dynamical Systems
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DOI：
10.1112/plms/s3-4.1.168
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发表时间：
1954
影响因子：
1.8
通讯作者：
Y. N. Dowker;F. G. Friedlander
中科院分区：
文献类型：
--
作者：
Y. N. Dowker;F. G. Friedlander
一个“DISCRETE动力系统”由一个紧致空间X和一个给定的一对一连续映射ToIX到自身的迭代所产生的X到自身的离散映射群组成。“连续动力系统”(流)由紧空间X和X到自身的可加单参数映射Tt(t是实数)群组成。在这类系统的研究中，点的余限集的概念具有一定的重要性。粗略地说，这是从点开始的正半轨迹的极限集；它是闭的，并且在T.GD Birkhoff(2)下不变，这一概念归因于T.GD Birkhoff(2)，他特别证明了在连续情况下，余极限集总是连通的。人们期望w-极限集具有更多的内在性质，因为它是以特殊的方式产生的，而且寻找表征余极限集的内在性质也是很自然的。这里定义了一个空间X称为T-连通的(关于给定的X到自身的一对一连续映射T)，如果它不包含映射到它自己内部的真闭子集。这一性质，类似于但不等价于区域递归，被证明为每个余限集所具有的，并反过来意味着X是第二动力系统的某一点的余限集，该第二动力系统包括{X，T)(定理I和定理II)。如果T是恒等式，则T-连通性退化为通常意义下的连通性。一般而言，T-连通性在某种程度上类似于连通性；下面的定理III说明了这一点，它将著名的Sierpinski[参见(5)]分解定理从连续集推广到T-连通集。
A'DISCRETE dynamical system'consists of a compact space X and the discrete group of mappings of X onto itself generated by the iteration of a given one-to-one continuous mapping ToiX onto itself. A'continuous dynamical system'(a flow) consists of a compact space X and an additive one-parameter group of mappings Tt (t a real number) of X onto itself. In the study of such systems the notion of the co-limit set of a point is of some importance. Roughly speaking, this is the limit set of the positive half-trajectory from the point; it is closed, and invariant under T. GD Birkhoff (2), to whom this notion is due, showed inter alia that, in the continuous case, an co-limit set is always connected. One would expect an w-limit set to have further intrinsic properties, because of the special way in which it is generated, and it is also natural to look for an intrinsic property which characterizes an co-limit set. In this note such a property is defined.A space X is here called T-connected (with respect to a given one-to-one continuous mapping T of X onto itself) if it contains no proper closed subset which is mapped into its own interior. This property, similar but not equivalent to regional recurrence, is shown to be possessed by every co-limit set, and in turn to imply that X is the co-limit set of some point of a second dynamical system which'includes'{X, T)(Theorems I and II). If T is the identity, T-connectedness reduces to connectedness in the usual sense. In general, T-connectedness is to some extent analogous to connectedness; this is illustrated by Theorem III below, which extends Sierpinski's well-known [cf.(5)] decomposition theorem from continua to T-connected sets.