  In this article, we explore both numerically and analytically how the
dynamical environments of mildly relativistic binaries evolve with increasing
the general relativity factor $\gamma$ (the normalized inverse of the binary
size measured in the units of the gravitational radius corresponding to the
total mass of the system). Analytically, we reveal a phenomenon of the
relativistic shifting of mean-motion resonances: on increasing $\gamma$, the
resonances between the test particle and the central binary shift, due to the
relativistic variation of the mean motions of the primary and secondary
binaries and the relativistic advance of the tertiary's pericenter. To exhibit
the circumbinary dynamics globally, we numerically integrate equations of the
circumbinary motion of a test particle, and construct relevant scans of the
maximum Lyapunov exponents and stability diagrams in the "pericentric distance
-- eccentricity" plane of initial conditions. In these scans and diagrams,
regular and chaotic domains are identified straightforwardly. Our analytical
and numerical estimates of the shift size are in a good agreement. Prospects
for identification of the revealed effect in astronomical observations are
discussed.
