Opuscula Math. 31, no. 3 (2011), 425-431
http://dx.doi.org/10.7494/OpMath.2011.31.3.425

Opuscula Mathematica

# A note on invariant measures

Piotr Niemiec

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.

Keywords: invariant measures, equicontinuous semigroups, compact spaces.

Mathematics Subject Classification: 28C10, 54H15.

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• Piotr Niemiec
• Jagiellonian University ,Institute of Mathematics, ul. Łojasiewicza 6, 30-348 Kraków, Poland
• Revised: 2010-10-29.
• Accepted: 2010-11-11.