Abstract
In this paper, the entropy of discrete Heisenberg group actions is considered. Let α be a discrete Heisenberg group action on a compact metric space X. Two types of entropies, \(\tilde{h}(\alpha)\) and h(α) are introduced, in which \(\tilde{h}(\alpha)\) is defined in Ruelle’s way and h(α) is defined via the natural extension of α. It is shown that when X is the torus and α is induced by integer matrices then \(\tilde{h}(\alpha)\) is zero and h(α) can be expressed via the eigenvalues of the matrices.
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Acknowledgements
The authors would like to thank Prof. Weisheng Wu for suggesting us to consider the entropy of the Heisenberg group actions.
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Supported by NSFC (Grant Nos. 12171400, 12126102)
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Zhang, Y., Zhu, Y.J. A Note on the Entropy for Heisenberg Group Actions on the Torus. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-3076-3
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DOI: https://doi.org/10.1007/s10114-024-3076-3