  We study the localization-delocalization transition of Floquet eigenstates in
a driven fermionic chain with an incommensurate Aubry-Andr\'{e} potential and a
hopping amplitude which is varied periodically in time. Our analysis shows the
presence of a mobility edge separating single-particle delocalized states from
localized and multifractal states in the Floquet spectrum. Such a mobility edge
does not have any counterpart in the static Aubry-Andr\'{e} model and exists
for a range of drive frequencies near the critical frequency at which the
transition occurs. The presence of the mobility edge is shown to leave a
distinct imprint on fermion transport in the driven chain; it also influences
the Shannon entropy and the survival probability of the fermions at long times.
In addition, we find the presence of CAT states in the Floquet spectrum with
weights centered around a few nearby sites of the chain. This is shown to be
tied to the flattening of Floquet bands over a range of quasienergies. We
support our numerical studies with a semi-analytic expression for the Floquet
Hamiltonian ($H_F$) computed within a Floquet perturbation theory. The
eigenspectra of the perturbative $H_F$ so obtained exhibit qualitatively
identical properties to the exact eigenstates of $H_F$ obtained numerically.
Our results thus constitute an analytic expression of a $H_F$ whose spectrum
supports multifractal and CAT states. We suggest experiments which can test our
theory.
