A note on invariant measures

The aim of the paper is to show that if \(\mathcal{F}\) is a family of continuous transformations of a nonempty compact Hausdorff space \(\Omega\), then there is no \(\mathcal{F}\)-invariant probabilistic Borel measures on \(\Omega\) iff there are \(\varphi_1,\ldots,\varphi_p \in \mathcal{F}\) (for some \(p \geq 2\)) and a continuous function \(u:\, \Omega^p \to \mathbb{R}\) such that \(\sum_{\sigma \in S_p} u(x_{\sigma(1)},\ldots ,x_{\sigma(p)}) = 0\) and \(\liminf_{n\to\infty} \frac1n \sum_{k=0}^{n-1} (u \circ \Phi^k)(x_1,\ldots,x_p) \geq 1\) for each \(x_1,\ldots,x_p \in \Omega\), where \(\Phi:\, \Omega^p \ni (x_1,\ldots,x_p) \mapsto (\varphi_1(x_1),\ldots,\varphi_p(x_p)) \in \Omega^p\) and \(\Phi^k\) is the \(k\)-th iterate of \(\Phi\). A modified version of this result in case the family \(\mathcal{F}\) generates an equicontinuous semigroup is proved