The topological trace formula is a Trace formula for Topological dynamics.

$\sum_{\alpha} \frac{z \lambda_\alpha}{1-z\lambda_\alpha}=\sum_p \frac{n_p t_p}{1-t_p}$

See here and here. Also here. Here $t_p = n_p$ if the prime cycle $p$ exists ($n_p$ being its length, and is $0$ otherwise.

http://www.chaosbook.org/course1/Course2w9.html

This formula has uses for deriving a formula for the Topological entropy