Why is the Ruelle zeta function called a zeta function?

Question

Let $f \colon X \to X$ be a topological dynamical system. Write $|\operatorname{Fix}(f^n)|$ for the number of fixed points of the $n$-fold composition of $f$. Then the Ruelle zeta function is defined as $$\zeta(s) = \exp \bigg( \sum_{n \geq 1} |\operatorname{Fix}(f^n)| \frac{s^n}{n}\bigg).$$ Contrast this with the Riemann zeta function, defined via the formula $$\zeta(s) = \sum_{n \geq 1} \frac{1}{n^{-s}}.$$ From my understanding, we call something a zeta function if it resembles the Riemann zeta function. But these two functions look absolutely nothing alike. How can I see the similarity? I presume I need to make some kind of clever substitution to compare them, but I don't see how.

Answer 1

Perhaps you can better see a connection from the product formulae for the two zeta functions.

For the Riemann zeta function, we have

$$\zeta(s) = \prod_{p\text{ prime}}\frac{1}{1-p^{-s}}$$

and for the Ruelle zeta function, we have

$$\zeta(s) = \prod_{\gamma \text{ periodic orbit}}\frac{1}{1-s^{|\gamma|}}$$

where $|\gamma|$ is the length of the orbit $\gamma$ (the prime period of its constituent points).

At least these formulae are much similar in form than the summation formulae, of course there is still a difference in that, for one, the variable is an exponent appearing in the denominator of the multiplicative terms, and in the other it just appears as the base of a power.