  Exact results are derived relating quantities computable by the so-called
damage spreading method and thermodynamic functions for the N-color
Ashkin-Teller model. The results are valid for any ergodic dynamics. Since we
restrict our analysis to the ferromagnetic case the results are also valid for
any translational invariant lattice. The derived relations should be used in
order to determine numerically the N-color Ashkin-Teller critical exponents
with better accuracy and less computational efforts than standard Monte Carlo
simulations.
