动力系统的斑逆极限--动力学、特征和局部拓扑
Plaque Inverse Limit of a Dynamical System - Dynamics, Signatures and Local Topology
摘要：
在\cite{CCG}中引入并研究了黎曼曲面分支覆盖自映射的Plaque逆极限。P.I.L.点$x$。被称为普通人在$x$处具有自然的黎曼曲面结构，否则被称为不规则。给出了$x$关于临界点$c$的签名$sign(x,c)$的概念，证明了它是P.I.L.被引进和发展。证明了$sign(x,c)$对某些临界点$c$是非平凡的当且仅当$x$是不规则点.结果表明，PIL的局部拓扑结构在不规则点处，$x$具有这样的性质，即从其任何邻域中移除$x$会将该邻域的某些路径连接分量分解成无数个路径连接分量。最后，计算了各种签名，包括超吸引和吸引圈的不变提升的签名以及抛物圈的不变提升的某些签名。所有这些签名都有一个极大元素。在这项工作中，我们表明，局部拓扑P.I.L.在不规则点上，不同类型的签名是不同的。证明了不规则点$x$的局部拓扑具有一个性质，即对$x$的任意邻域$V$和$V$中的某点$y\ne x$，开集$V-\{y\}$由不可数个路连通分支组成，当且仅当对某个临界点$c$，签名$sign(x,c)$没有极大元.其次，对于一个多项式函数，我们计算了一个抛物循环关于某个常返临界点的不变提升的签名。与\cite{CCG}中研究的情况不同，该签名没有最大元素。我们表明，所有其他的不规则点，除了不变电梯的超吸引，吸引，抛物循环，有没有最大元素的一些经常性的临界点。
英文摘要：
The Plaque Inverse Limit of a branched covering self-map of a Riemann surface was introduced and studied in \cite{CCG}. A point $x$ of P.I.L. was called regular if P.I.L. has the natural Riemann Surface structure at $x$ and was called irregular otherwise. The notion of the signature $sign(x,c)$ of $x$ with respect to a critical point $c$, which was shown to be a local invariant of P.I.L. was introduced and developed. It was shown that $sign(x,c)$ is nontrivial for some critical points $c$ if and only if $x$ is an irregular point. It was shown that the local topology of P.I.L. at an irregular point $x$ has a property, that removing $x$ from any its neighborhood breaks some path-connected component of that neighborhood into an uncountable number of path-connected components. Finally, various signatures, including signatures of the invariant lifts of super-attracting and attracting cycles and certain signatures of the invariant lift of a parabolic cycle, were computed. All these signatures had a maximal element. In this work we show that the local topology of P.I.L. at irregular points with different types of signatures is different. Namely, we prove that the local topology at an irregular point $x$ has a property, that for any neighborhood $V$ of $x$ and for some point $y\ne x$ in $V$, the open set $V-\{y\}$ consists of uncountable number of path-connected components, if and only if the signature $sign(x,c)$, for some critical point $c$, has no maximal element. Next, for a polynomial functions, we compute the signature of the invariant lift of a parabolic cycle with respect to a certain recurrent critical point. This signature, unlike the cases studied in \cite{CCG}, has no maximal element. We show that all other irregular points, except the invariant lifts of super-attracting, attracting, and parabolic cycles, have no maximal element with respect to some recurrent critical point.
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