  The interplay between Floquet driving and non-Hermitian gain/loss could give
rise to intriguing phenomena including topological funneling of light,
edge-state delocalization, anomalous topological transitions and Floquet
non-Hermitian skin effects. In this work, we uncover two unique phenomena in
Floquet systems caused by gain and loss. First, multiple topological
transitions from anomalous Floquet second-order topological insulators to
anomalous Floquet first-order topological insulators and then to normal
insulators can be induced by gain and loss. Interestingly, the resulting
anomalous Floquet insulators further carry hybrid skin-topological boundary
modes, which could either be fully localized or localized to different edges at
different time slices and traversing along all edges in a single driving
period. The topological phase transitions are also shown to be detectable
through studies of transmission properties in the setting of coupled ring
resonators. Second, gain and loss are found to induce singularities in the
Floquet spectral, around which anomalous transmissions at flat quasienergy
bands are predicted. These discoveries not only enhanced our understanding of
topological matter and phase transitions in driven non-Hermitian systems, but
also promoted their experimental realizations in optical and acoustic settings.
