2015/06/29 by Richard Miles, Miles, Richard
Chemistry · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Molecular spectroscopy and chirality #Origins and Evolution of Life #math.DS
paper · pdf · doi:10.48550/arxiv.1506.08555
openalex publication_date 2015/06/29 · arxiv created 2015/10/30 · arxiv updated 2015/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 2, 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.