Periodic points and non-wandering points of continuous dynamical systems (Q1265470)

Let \(I\) be an interval in \(\mathbb{R}\), that is \(I\subset\mathbb{R}\). The author proves that if \(\psi:\mathbb{R}_+\times I\to I\) is a continuous semiflow on \(I\), the existence of a periodic orbit implies that \(\psi\) is the identity; or equivalently, the only compact orbits are reduced to the fixed point of \(\psi\). In the case, when \(I\) is replaced by \(S^1\)-circle, the existence of periodic points implies that \(\psi\) is a periodic flow. To obtain similar conclusions for flows and semiflows in higher dimensions, some additional conditions should be imposed on \(\psi\). For example, in the case of complex dimension one, i.e., when \(\psi\) acts holomorphically on the open unit disk of the complex plane, the periodicity of \(\psi\) follows from the fact that the set of non-wandering points of \(\psi\) is not empty.