A survey of orbit Dirichlet series (Q2905842)

For a continuous mapping \(T: X\to X\), where \(X\) is a compact metric space the number of points of period \(n\) is defined by \(F_n(T) =\#\{x\in X\mid T_n(x) = x\},\) and then \(F_n(T) =\sum_{d\mid n} dO_n(T)\), where \(O_n(T)\) is the number of closed orbits of length \(n\) for any \(n\geq 1\). NEWLINENEWLINEThe Möbius inversion formula gives \(O_n(T) =\frac 1n \sum_{d\mid n}\mu({n\over d}) F_d(T)\), where \(\mu\) is the Möbius function.NEWLINENEWLINERecently, the author and \textit{T. B. Ward} [J. Integer Seq. 12, No. 2, Article ID 09.2.4, 20 p. (2009; Zbl 1254.37020)] introduced the orbit Dirichlet series \(d_T(s) =\sum_{n=1}^\infty \frac{O_n(T)}{n^s}\) and developed analytic tools to study orbit growth using properties of this function.NEWLINENEWLINEIn the present paper the author surveys the results of the paper cited above and considers the Cartesian product of closed orbits and orbit Dirichlet series for these maps. Finally, the theorem for abscissa of convergence of the Cartesian product of orbit Dirichlet series is given.