Shadowing vs. distality for actions of ℝⁿ

Abstract

This article introduces the notion of weakly parametrized (wp) shadowing for actions of groups ℤ ^m  × ℝ ⁿ , where m, n ≥ 0 and m + n > 0. The possibility of coexistence of distality and shadowing for actions of ℝ ⁿ is discussed. It is proven that an equicontinuous action of ℝ ⁿ on a compact connected space possessing wp-shadowing is actually minimal. Moreover, distal real flows (ℝ-actions) on one-dimensional compact metric spaces are characterized as constant-one suspensions over adding machines.

Keywords:

• dynamical system
• topological group
• distality
• shadowing
• pseudo-orbit

AMS Subject Classifications:

• 37C50
• 22F05

Acknowledgements

The authors are grateful to the referee for many helpful suggestions that greatly improved the clarity of this article. Research leading to this article has been fully supported by the KBN grant NN201270035. The authors also express their gratitude to Dr Krzysztof Ciesielski for his helpful comments and advice on the choice of terminology.