A compact space X is said to be minimal if there exists a map \(f:X\rightarrow X\) such that the forward orbit of any point is dense in X. 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\(f:X\rightarrow X\)使得任何点的前向轨道在X 中是稠密的，则称紧致空间X是最小的. 我们考虑刚性最小空间，受 Downarowicz、Snoha 和 Tywoniuk (J Dyn Differ Equ, 29:243–257, 2017) 对具有由最小同胚生成的循环同胚群的空间的最新结果以及第一作者的结果的启发， Clark 和 Oprocha (Adv Math, 335:261–275, 2018) 在空间中每个同胚的平方都是相同最小同胚的幂。我们证明这两个类不重合，这产生了一类新的空间，它承认最小同胚，但没有最小映射。我们修改后一类例子以首次展示具有退化同胚群的极小空间的存在。最后，我们给出了一种用最小同胚生成的循环同胚群构造可分解紧连通空间的方法，