A Note on Invariant Measures
نویسنده
چکیده
The aim of the paper is to show that if F is a family of continuous transformations of a nonempty compact Hausdorff space Ω, then there is no F-invariant probabilistic regular Borel measures on Ω iff there are φ1, . . . , φp ∈ F (for some p ≥ 2) and a continuous function u : Ω → R such that P σ∈Sp u(xσ(1), . . . , xσ(p)) = 0 and lim infn→∞ 1 n Pn−1 k=0 (u ◦ Φ )(x1, . . . , xp) ≥ 1 for each x1, . . . , xp ∈ Ω, where Φ: Ω 3 (x1, . . . , xp) 7→ (φ1(x1), . . . , φp(xp)) ∈ Ω and Φ is the k-th iterate of Φ. A modified version of this result in case the family F generates an equicontinuous semigroup is proved.
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