Abstract
This article introduces and investigates the basic features of a dynamical zeta function for group actions, motivated by the classical dynamical zeta function of a single transformation. A product formula for the dynamical zeta function is established that highlights a crucial link between this function and the zeta function of the acting group. A variety of examples are explored, with a particular focus on full shifts and closely related variants. Amongst the examples, it is shown that there are infinitely many non-isomorphic virtually cyclic groups for which the full shift has a rational zeta function. In contrast, it is shown that when the acting group has Hirsch length at least two, a dynamical zeta function with a natural boundary is more typical. The relevance of the dynamical zeta function in questions of orbit growth is also considered.
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Communicated by S. G. Dani.
This work was initiated with the support of the London Mathematical Society Scheme IV grant number 41352, which the author gratefully acknowledges.
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Miles, R. A dynamical zeta function for group actions. Monatsh Math 182, 683–708 (2017). https://doi.org/10.1007/s00605-016-0909-x
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DOI: https://doi.org/10.1007/s00605-016-0909-x