Chris Hillman, in a 23/10/01 posting to sci.physics.research wrote:

"One of the reasons why dynamical zeta functions and the associated transfer operators are so important in dynamical systems is that if you know enough about the location of the zeros/poles, you can estimate the rate of decay of correlations, i.e. the mixing ("randomization") rate; see for example the book

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, Vol. 16. World Scientific, 2000

Note that these ideas relate number theory, exactly solvable models (Yang-Baxter, etc.), and symbolic dynamics. It never hurts to repeat that the famous Riemann zeta function is nothing other than the dynamical zeta function of the symbolic dynamical system ("full two shift") obtained by equipping the set X of all binary sequences with the metric defined by d(x,y) = 1/2ⁿ where x,y are Z-indexed sequences whose closest discrepancy to the index 0 occurs at the index +n. This makes (X,d) into a compact metric space which has the character of a "Cantor dust"."
 

Mark Pollicott, when the highlighted section was brought to his attention, commented:

"I am afraid I didn't quite understand what Chris Hillman was hinting at. The "dynamical zeta function" associated with the basic full shift on 2 symbols is the usual Artin-Mazur zeta function:

z -> exp(\sum_n z^n [no. periodic points of period n]/n)

The Ruelle zeta function allows you to take a function f, say, on the shift and weight an orbit Tⁿx = x by the sum of the values of f around the orbit. This allows one to slip in a extra variable s, say:

zeta(z,s) = exp(\sum_n z^n \sum_{T^nx=x} e^{-s[f(x)+f(Tx)+ ...+f(T^{n-1}x)]})

The function f allows a bit of freedom. I don't know any examples of f where zeta(1,s) is the Riemann zeta function. In fact, because there is an extension of zeta(1,s) to a strip, and it cannot be zero, any such example would prove more than is currently known about the Riemann Zeta function.

Sometimes, the Riemann Zeta function occurs for other zeta functions (Gutzwiller, if I remember correctly, gets it for the spectral zeta function of a punctured torus) or as part of a more general expression (Mayer's results on the Selberg Zeta function for the modular group).