A dynamical zeta function for group actions

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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Correspondence to Richard Miles.

Additional information

This work was initiated with the support of the London Mathematical Society Scheme IV grant number 41352, which the author gratefully acknowledges.

Communicated by S. G. Dani.

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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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Keywords

  • Dynamical zeta function
  • Group action
  • Periodic orbit
  • Product formula
  • Natural boundary
  • Full shift

Mathematics Subject Classification

  • Primary: 37A45
  • 11M41
  • 37A35
  • 37C30
  • Secondary: 37C25
  • 37C35
  • 37C85
  • 22F05