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)