On a symbolic dynamical zeta function and applications (Q837531)

Let \(A\) be a countable set and \(A^{\mathbb{N}*}\) be the set of infinite sequences \((i_1,i_2,\ldots)\) endowed with the product topology. Also, let \(\mathcal{F}\) be a closed subset of \(A^{\mathbb{N}*}\) which is invariant under the shift map \(T(i_1,i_2,\ldots) = (i_2,\ldots)\). The set of periodic points under \(T\) of period \(n\) is denoted by \(\text{Fix}(T)\). For \(\varphi: \mathcal{F} \to \mathbb{R}\) fulfilling certain assumptions the author defines the zeta function \[ \zeta(\varphi) := \sum_{n \geq 1} \frac{1}{n} \, \zeta_n(\varphi) \] with \[ \zeta_n(\varphi) := \inf_{R > 0} \sum_{\eta \in \text{Fix}(T^n) \atop \eta_n > R} e^{\varphi_n(\eta)} \quad \text{and} \quad \varphi_n(\eta):= \sum_{k=0}^{n-1} \varphi(T^k \eta). \] The author's main result is that the above zeta function \(\zeta(\varphi)\) is a meromorphic function on the function space \[ \left\{ \varphi: \; P_\infty(\varphi):= \limsup_{n \to \infty} \frac{1}{n} \log \left( \limsup_{j \to \infty} \sum_{i_1,\i_2, \ldots, i_n} e^{\varphi_n(i_1,i_2, \ldots, i_n,j)} \right)\, < \, 0 \right\}. \] The zeta function has a pole if and only if \(P_\infty(\varphi)=0\). For fixed \(\varphi\) the author continues to discuss the analytic and meromorphic properties of the zeta function \(\zeta(z) := \zeta(\varphi + \log z)\), \(z \in \mathbb{C}\). Applications to interval maps and suspensions are given.