  We study the many-body localization (MBL) transition of Floquet eigenstates
in a driven, interacting fermionic chain with an incommensurate Aubry-Andr\'{e}
potential and a time-periodic hopping amplitude as a function of the drive
frequency $\omega_D$ using exact diagonalization (ED). We find that the nature
of the Floquet eigenstates change from ergodic to Floquet-MBL with increasing
frequency; moreover, for a significant range of intermediate $\omega_D$, the
Floquet eigenstates exhibit non-trivial fractal dimensions. We find a possible
transition from the ergodic to this multifractal phase followed by a gradual
crossover to the MBL phase as the drive frequency is increased. We also study
the fermion auto-correlation function, entanglement entropy, normalized
participation ratio (NPR), fermion transport and the inverse participation
ratio (IPR) as a function of $\omega_D$. We show that the auto-correlation,
fermion transport and NPR displays qualitatively different characteristics
(compared to their behavior in the ergodic and MBL regions) for the range of
$\omega_D$ which supports multifractal eigenstates. In contrast, the
entanglement growth in this frequency range tend to have similar features as in
the MBL regime; its rate of growth is controlled by $\omega_D$. Our analysis
thus indicates that the multifractal nature of Floquet-MBL eigenstates can be
detected by studying auto-correlation function and fermionic transport of these
driven chains. We support our numerical results with a semi-analytic expression
of the Floquet Hamiltonian obtained using Floquet perturbation theory (FPT) and
discuss possible experiments which can test our predictions.
