A note on invariant measures

Abstract

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.