Maps of the interval with closed periodic set
Maps of the interval with closed periodic set
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DOI：
10.1090/s0002-9939-1982-0656122-2
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发表时间：
1982-03
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通讯作者：
Z. Nitecki
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作者：
Z. Nitecki
我们证明，对于区间上任何其周期点构成闭集的连续映射，每个非游荡点都是周期的，且最小周期是2的幂次方。布洛克（Block）表明[Bi]，如果区间上的连续映射$f:I\to I$的周期点集$Per(f)$是有限的且仅由不动点组成，那么非游荡集$\Omega(f)$等于$Per(f)$。科文（Coven）和赫德伦（Hedlund）[CH]扩展了这一结果，从较弱的假设即$f$的某个幂次$g = f^{n_1}$同时固定所有周期点得出了相同的结论。与这些相关的其他结果在[B2, CH, L]中得以确立。在本文中，我们扩展上述结果。
定理：如果$f:I\to I$是连续的且$Per(f)$是闭集，那么$\Omega(f)=Per(f)$。
我要感谢伊桑·科文和路易斯·布洛克就这个问题进行的有益讨论，包括我原始版本中的一个漏洞。我最近听说熊金城[X]对上述定理有一个独立的证明。我们注意到$Per(f)$是闭集并不意味着最小周期集是有限的，所以科文 - 赫德伦的结果不一定适用。为了构造一个例子，我们只需将映射$f_n:[0,1]\to[0,n]$连接起来，使得$f_n(k)=k$且$Per(f_n)$包含最小周期为$2^n$的点，但没有更高周期的点。从我们定理的证明中我们将看到，这个例子本质上是$Per(f)$是闭集且[CH]不适用的唯一情况。
我们的出发点是布洛克的同宿点定理[B3]。给定$f$的一个周期点$p$以及$f$的一个幂次$g = f^{n}$，通过以下方式定义$p$的$g$ - 轨道的全、左和右不稳定集：
\(\omega(p,g)=\bigcap_{\epsilon>0}\bigcup_{k>0}g^{k}(p-\epsilon,p+\epsilon)\)
\(W^U(p,g,L)=\bigcap_{\epsilon>0}\bigcup_{k>0}g^{k}(p-\epsilon,p]\)
\(W^U(p,g,R)=\bigcap_{\epsilon>0}\bigcup_{k>0}g^{k}[p,p+\epsilon)\)
很明显，当$g(p)=p$时，每个不稳定集都是一个包含$p$的区间（也许没有其他点）。可以看出\(W^U(p,g)=W^U(p,g,L)\cup W^U(p,g,R)\)。
1981年8月25日收到编辑来稿。
1980年数学学科分类。主要58F20，54H20。
We 8how that for any continuou8 map of the interval who8e periodic point8 form a clo8ed 8et, every nonwandering point i8 periodic with least period a power of two. Block showed [Bi] that if the set Per(f) of periodic points for a continuous map of the interval f: I -I is finite and consists only of fixed points, then the nonwandering set Ql(f) equals Per(f). Coven and Hedlund [CH] extended this, obtaining the same conclusion from the weaker hypothesis that some power g = fnl of f simultaneously fixes all the periodic points. Other results related to these are established in [B2, CH, L]. In this paper, we extend the results stated above. THEOREM. If f: I -I is continuous and Per(f) is a closed set, then Q(f) Per(f). I would like to thank Ethan Coven and Louis Block for useful conversations about this problem, including a gap in my original version. I have heard recently of an independent proof of the theorem above by Jin-Cheng Xiong [X]. We note that Per(f) closed does not imply that the set of least periods is finite, so that Coven-Hedlund's result need not apply. To construct an example, we simply string together maps fn: [, 1 ] -+ [ -n ] so that fn(k) = k and Per(f,) contains points of least period 2n, but none higher. We will see from the proof of our theorem that this example is in essence the only situation in which Per(f) is closed and [CH] does not apply. Our point of departure is Block's homoclinic point theorem [B3]. Given a periodic point p for f and a power of f, g = fn, define the full, left, and right unstable sets of the g-orbit of p by w(p, g) n u gk (p ,p +, E>O k>O (1) WW(p,g,L) nf U gk(p_,p], E>O k>O WU(p,g,R)= n U gk[p,pc) E>O k>O It is clear that when g(p) = p, each unstable set is an interval containing p (and perhaps nothing else). One can see that WU(p, g) = WU(p, g, L) U WU(p, g, R). Received by the editors August 25, 1981. 1980 Mathematics Subject Classification. Primary 58F20, 54H20.