A dynamical zeta function for group actions

MILES, Richard (2017). A dynamical zeta function for group actions. Monatshefte für Mathematik, 182 (3), 683-708.

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Official URL: https://link.springer.com/article/10.1007/s00605-0...
Link to published version:: https://doi.org/10.1007/s00605-016-0909-x
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    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.

    Item Type: Article
    Departments - Does NOT include content added after October 2018: Faculty of Social Sciences and Humanities > Department of Teacher Education
    Identification Number: https://doi.org/10.1007/s00605-016-0909-x
    Page Range: 683-708
    Depositing User: Richard Miles
    Date Deposited: 21 Dec 2017 12:36
    Last Modified: 18 Mar 2021 16:31
    URI: https://shura.shu.ac.uk/id/eprint/17217

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