  We analyze the dynamics of a quantum kicked rotor (QKR) driven with a binary
Fibonacci sequence of two distinct drive amplitudes. While the dynamics at low
drive frequencies is found to be diffusive, a long-lived pre-ergodic regime
emerges in the other limit. Further, the dynamics in this pre-ergodic regime
can be associated with the onset of a dynamical quasi-localization, similar to
the dynamical localization observed in a regular QKR. We establish that this
peculiar behavior arises due to the presence of localized eigenstates of an
approximately conserved effective Hamiltonian, which drives the evolution at
Fibonacci instants. However, the effective Hamiltonian picture does not persist
indefinitely and the dynamics eventually becomes ergodic after asymptotically
long times.
