紧度量空间上单边动力系统的Takens型重构定理
Takens-type reconstruction theorems of one-sided dynamical systems on compact metric spaces
摘要：
重构定理处理由紧度量空间$X$的映射$T:X\to X$以及从$X$到实直线$\R$的可观测$f:X \to \R$给出的动力系统。1981年，Takens利用Whitney嵌入定理证明了：如果$T:M\to M$是紧致光滑流形$M$上的一个带$\dim M=d$的自同态，则对于一般的$(T,f)$，元素$x \in M$和相应的序列$(fT^j(x))_{j=0}^{2d}$之间存在一个双射; 2002年，Takens证明了自同态的一个推广版本。在自然科学和物理工程中，分形集和更复杂的空间的重要性有所增加，在数学中，也研究了这些空间的许多拓扑和动力学性质以及随机分析。本文利用拓扑学方法，将紧致光滑流形的Takens重构定理推广到一类紧致度量空间的单边动力系统的重构定理，这类空间包含PL-流形、分支流形和一些分形集，如Menger流形、Sierpienski地毯、Sierpienski垫片和枝晶等.
英文摘要：
The reconstruction theorem deals with dynamical systems that are given by a map $T:X\to X$ of a compact metric space $X$ together with an observable $f:X \to \R$ from $X$ to the real line $\R$. In 1981, by use of Whitney's embedding theorem, Takens proved that if $T:M\to M$ is a diffeomorphism on a compact smooth manifold $M$ with $\dim M=d$, for generic $(T,f)$ there is a bijection between elements $x \in M$ and corresponding sequence $(fT^j(x))_{j=0}^{2d}$, and moreover, in 2002 Takens proved a generalized version for endomorphisms. In natural sciences and physical engineering, there has been an increase in importance of fractal sets and more complicated spaces, and also in mathematics, many topological and dynamical properties and stochastic analysis of such spaces have been studied. In the present paper, by use of some topological methods we extend the Takens' reconstruction theorems of compact smooth manifolds to reconstruction theorems of one-sided dynamical systems for a large class of compact metric spaces, which contains PL-manifolds, branched manifolds and some fractal sets, e.g. Menger manifolds, Sierpiński carpet and Sierpiński gasket and dendrites, etc.
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