Noninvertible minimal maps
Noninvertible minimal maps
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DOI：
10.4064/fm168-2-5
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发表时间：
2001-01-01
影响因子：
0.6
通讯作者：
Trofimchuk, S
中科院分区：
文献类型：
--
作者：
Kolyada, S;Snoha, L;Trofimchuk, S
对于由一个紧豪斯多夫空间\(X\)以及\(X\)的一个连续自映射\(f\)所给出的一个离散动力系统，研究了\(f\)的极小性、可逆性和开放性之间的联系。证明了任何极小映射都是弱开的，即把开集映射到内部非空的集合（并且如果它是开的，那么它是一个同胚）。进一步证明了如果\(f\)是极小的且\(A\subseteq X\)，那么\(f(A)\)和\(f^{-1}(A)\)与\(A\)都具有那些描述集合大小的拓扑性质。利用这些结果证明了在一个紧度量空间中的任何极小映射几乎是一一的，而且当限制在一个合适的不变剩余集上时它变成一个极小同胚。最后，给出了环面上两种不可逆极小映射的例子——这些是作为环面的一个适当极小同胚的因子或者扩张而得到的。
For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A subset of or equal to X then both f(A) and f(-1)(A) share with A those topological properties which describe how large a set is. Using these results it is proved that any minimal map in a compact metric space is almost one-to-one and, moreover, when restricted to a suitable invariant residual set it becomes a minimal homeomorphism. Finally, two kinds of examples of noninvertible minimal maps on the torus are given-these are obtained either as a factor or as an extension of an appropriate minimal homeomorphism of the torus.