  We study the Potts model on a rectangular lattice with aperiodic modulations
in its interactions along one direction. Numerical results are obtained using
the Wolff algorithm and for many lattice sizes, allowing for a finite-size
scaling analyses to be carried out. Three different self-dual aperiodic
sequences are employed, which leads to more precise results, since the exact
critical temperature is known. We analyze two models, with six and fifteen
number of states: both present first-order transitions on their uniform
versions. We show that the Harris-Luck criterion, originally introduced in the
study of continuous transitions, is obeyed also for first-order ones. Also, we
show that the new universality class that emerges for relevant aperiodic
modulations depends on the number of states of the Potts model, as obtained
elsewhere for random disorder, and on the aperiodic sequence. We determine the
occurrence of log-periodic behavior, as expected for models with aperiodic
modulated interactions.
