  We investigate the dynamical critical behavior of the two-dimensional
three-state Potts model with single spin-flip dynamics in equilibrium. We focus
on the mean-squared deviation of the magnetization $M$ (MSD$_{M}$) as a
function of time, as well as on the autocorrelation function of $M$. Our
simulations reveal the existence of two crossover behaviors at times $\tau_1
\sim L^{z_1}$ and $\tau_2 \sim L^{z_2}$, separating three dynamical regimes.
MSD$_{M}$ appears to shift from ordinary diffusion in the first regime, to
anomalous diffusion in the second, and finally to be constant in the third
regime. The magnetization autocorrelation function on the other hand is found
to fluctuate between exponential decay, stretched-exponential decay, and then
again exponential decay along these three regimes. This behavior is in
agreement with the one reported recently for the two-dimensional Ising
ferromagnet [Phys. Rev. E {\bf 108}, 034118 (2023)], indicating that the
existence of two dynamic critical exponents is not a peculiarity of the Ising
model itself. A comparison of both MSD$_{M}$ and the magnetization's
autocorrelation function suggests that within our numerical accuracy the
exponents $z_1$ and $z_2$ are shared between the Ising and three-state Potts
models at least for the particular case of single spin-flip dynamics studied
here, even though their equilibrium universality classes are clearly distinct.
Continuity of MSD$_{M}$ requires that $\alpha (z_2 - z_1) = \gamma/\nu - z_1$,
in which $\alpha$ is the anomalous exponent in the intermediate regime. Since
the ratio $\gamma/\nu$ is not shared between the two models, it follows that
$\alpha$ is not shared either, an aspect well verified in our simulations.
Finally, we also discuss the relevance of our main findings using another
useful observable, namely the line magnetization $M_{l}$.
