  The existence of gapless boundary states is a key attribute of any
topological insulator. Topological band theory predicts that these states are
robust against static perturbations that preserve the relevant symmetries. In
this article, using Floquet theory, we examine how chiral symmetry-protection
extends also to states subject to time-periodic perturbations $-$ in
one-dimensional Floquet topological insulators as well as in ordinary
one-dimensional time-independent topological insulators. It is found that, in
the case of the latter, the edge modes are resistant to a much larger class of
time-periodic symmetry-preserving perturbations than in Floquet topological
insulators. Notably, boundary states in chiral time-independent topological
insulators also exhibit an unexpected resilience against a certain type of
symmetry-breaking time-periodic perturbations. We argue that this is a generic
property for topological phases protected by chiral symmetry. Implications for
experiments are discussed.
