On the existence of ergodic automorphisms in ergodic ${\mathbb Z} ^d$-actions on compact groups

Details On the existence of ergodic automorphisms in ergodic ${\mathbb Z} ^d$-actions on compact groups On the existence of ergodic automorphisms in ergodic ${\mathbb Z} ^d$-actions on compact groups

2009-04-06

arxiv.org/abs/0904.0848v1

Mathematics

Dynamical Systems

Scientific paper

Let $K$ be a compact metrizable group and $\Ga$ be a finitely generated group of commuting automorphisms of $K$. We show that ergodicity of $\Ga$ implies $\Ga$ contains ergodic automorphisms if center of the action, $Z(\Ga) = \{\ap \in {\rm Aut}(K) \mid \ap {\rm commutes with elements of \rm} \Ga \}$ has DCC. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which in particular, retrieves Theorems of Berend \cite{Be} and Schmidt \cite{Sc1} proved in this context.

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