For  a self map $f$ on a  topological space $X$ we replace "trace" with "determinant" in the  alternative Lefschetz formula $$\Lambda(f)=\sum(-1)^i trace(f^*)|H^i(X,\mathbb{Q})$$

So  we have 
 $$\Lambda'(f)=\sum(-1)^i Det(f^*)|H^i(X,\mathbb{Q})$$

What kind of  dynamical information we get with this invariant?(This  invariant or  any other invariant by replacing trace  with some  other invariant polynomials,i.e. the  coefficients  of  characteristic polynomials)