Localizing with respect to self-maps of the circle

We describe a general procedure to construct idempotent functors on the pointed homotopy category of connected ${\text{CW}}$-complexes, some of which extend $P$-localization of nilpotent spaces, at a set of primes $P$. We focus our attention on one such functor, whose local objects are ${\text{CW}}$-complexes $X$ for which the $p$th power map on the loop space $\Omega X$ is a self-homotopy equivalence if $p \notin P$. We study its algebraic properties, its behaviour on certain spaces, and its relation with other functors such as Bousfield's homology localization, Bousfield-Kan completion, and Quillen's plus-construction.