Boroński Jan P, Činč Jernej, Foryś-Krawiec Magdalena
Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland.
National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. Dubna 22, 70103 Ostrava, Czech Republic.
J Dyn Differ Equ. 2021;33(2):1023-1034. doi: 10.1007/s10884-020-09845-4. Epub 2020 Apr 1.
A compact space is said to be minimal if there exists a map such that the forward orbit of any point is dense in . We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha and Tywoniuk (J Dyn Differ Equ, 29:243-257, 2017) on spaces with cyclic group of homeomorphisms generated by a minimal homeomorphism, and results of the first author, Clark and Oprocha (Adv Math, 335:261-275, 2018) on spaces in which the square of every homeomorphism is a power of the same minimal homeomorphism. We show that the two classes do not coincide, which gives rise to a new class of spaces that admit minimal homeomorphisms, but no minimal maps. We modify the latter class of examples to show for the first time existence of minimal spaces with degenerate homeomorphism groups. Finally, we give a method of constructing decomposable compact and connected spaces with cyclic group of homeomorphisms, generated by a minimal homeomorphism, answering a question in Downarowicz et al.
如果存在一个映射，使得任何点的正向轨道在某个空间中是稠密的，那么这个紧致空间就被称为极小的。我们考虑刚性极小空间，这是受Dowarowicz、Snoha和Tywoniuk（《动力差分方程杂志》，29:243 - 257，2017）关于由极小同胚生成的同胚循环群的空间的近期结果，以及第一作者、Clark和Oprocha（《高等数学》，335:261 - 275，2018）关于每个同胚的平方是同一个极小同胚的幂的空间的结果的启发。我们证明这两类空间不重合，这就产生了一类新的空间，这类空间允许极小同胚，但不允许极小映射。我们修改后一类例子，首次展示了具有退化同胚群的极小空间的存在性。最后，我们给出一种构造由极小同胚生成同胚循环群的可分解紧致连通空间的方法，回答了Dowarowicz等人提出的一个问题。
