Artin–Mazur zeta function

Remarks

Example: let $f(z) = z^2$ on the unit circle in $\C$. The fixed points of $f^n$ are the solutions of $z^{2^n} = z$ with $\abs{z} = 1$, that is, the $(2^n - 1)$-th roots of unity, so $N_n = 2^n - 1$. Then $$\zeta_f(z) = \exp\left( \sum_{n \ge 1} \frac{(2z)^n}{n} - \sum_{n \ge 1} \frac{z^n}{n} \right) = \frac{1 - z}{1 - 2z}.$$

Non-example: the identity map of the circle has infinitely many fixed points, so $N_1$ is not finite and the definition does not apply.

If $N_n \le C a^n$ for constants $C$ and $a$, the series has a positive radius of convergence and $\zeta_f$ is a holomorphic function near $0$. For the shift on a subshift of finite type with transition matrix $A$, $N_n = \tr(A^n)$ and $\zeta_f(z) = 1 / \det(I - zA)$. For an Axiom A diffeomorphism of a compact manifold, the zeta function is rational (Manning). The function has the product form $\zeta_f(z) = \prod_{\gamma} (1 - z^{\abs{\gamma}})^{-1}$, a product over the periodic orbits $\gamma$ of $f$ with $\abs{\gamma}$ the number of points of $\gamma$.