  We study $p$-wave superconducting quasiperiodic chains with staggered
potential. The result shows a counter-intuitive phase transition phenomenon,
i.e., recurrent extension phase transition (REPT). By analyzing the
participation ration and scaling behavior, we prove the existence of REPT
phenomenon, which, in concrete terms, means that the system will repeatedly
return from the intermediate phase to the extended phase as the quasiperiodic
or staggered strength grows. Furthermore, our finding is also quite different
from the traditional understanding of intermediate phase (composed only of the
pure extended phase and pure localized phase) in that, the new intermediate
phase described here, stemming from the competition between staggered potential
and $p$-wave pairing, actually falls into three types by bringing in the
critical phase. To be specific, the new intermediate phases are composed of the
critical + extended states, the critical + localized states, and the critical +
extended + localized states, respectively.
