On Mixing and Ergodicity in Locally Compact Motion Groups

Details On Mixing and Ergodicity in Locally Compact Motion Groups On Mixing and Ergodicity in Locally Compact Motion Groups

2006-12-10

arxiv.org/abs/math/0612262v1

Journal fuer die reine und angewandte Mathematik, 625 (2008), 1--28.

Mathematics

Functional Analysis

Scientific paper

Let $G$ be a semi-direct product $G=A\times_\phi K$ with $A$ Abelian and $K$ compact. We characterize spread-out probability measures on $G$ that are mixing by convolutions by means of their Fourier transforms. A key tool is a spectral radius formula for the Fourier transform of a regular Borel measure on $G$ that we develop, and which is analogous to the well-known Beurling--Gelfand spectral radius formula. For spread-out probability measures on $G$, we also characterize ergodicity by means of the Fourier transform of the measure. Finally, we show that spread-out probability measures on such groups are mixing if and only if they are weakly mixing.

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