On rigid minimal spaces (Q2025687)

For compact \(X\) a map \(f:X\to X\) is minimal if the forward \(f\)-orbit of \(x\) is dense in \(X\) for each \(x\in X\); when such \(f\) exists \(X\) is called minimal. A nondegenerate compact space \(X\) is Slovak if its homeomorphism group \(\mathcal H(X)\) is cyclic on one generator \(h\) that is minimal; and is almost Slovak if \(\mathcal H(X)=\mathcal H_+(X)\cup\mathcal H_-(X)\) where \(\mathcal H_+(X)\) is as before and \(g^2\in\mathcal H_+(X)\) for each \(g\in\mathcal H_-(X)\). Examples of almost Slovak spaces that are not Slovak (these examples do not admit minimal non-invertible maps) and of minimal spaces having trivial homeomorphism group are given and discussed. These examples can be assumed to be decomposable continua.