Abstract
In this paper, it is shown that for d ∈ ℕ, a minimal system (X, T) is a d-step pro-nilsystem if its enveloping semigroup is a d-step top-nilpotent group, answering an open question by Donoso. Thus, combining the previous result of Donoso, it turns out that a minimal system (X, T) is a d-step pro-nilsystem if and only if its enveloping semigroup is a d-step top-nilpotent group.
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Acknowledgments
The authors would like to thank Professors W. Huang, S. Shao, X. Ye and Dr. F. Cai for helpful discussions and remarks. We thank Professor V. Bergelson for bringing our attention to this problem again. The authors were supported by NNSF of China (11431012).
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Qiu, J., Zhao, J. Top-nilpotent enveloping semigroups and pro-nilsystems. JAMA 148, 595–636 (2022). https://doi.org/10.1007/s11854-022-0238-z
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DOI: https://doi.org/10.1007/s11854-022-0238-z