A Note on the Entropy for Heisenberg Group Actions on the Torus

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, \(\tilde{h}(\alpha)\) $$\tilde{h}(\alpha )$$ and h(α) are introduced, in which \(\tilde{h}(\alpha)\) $$\tilde{h}(\alpha )$$ 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 \(\tilde{h}(\alpha)\) $$\tilde{h}(\alpha )$$ is zero and h(α) can be expressed via the eigenvalues of the matrices.