  It is shown theoretically that the renormalization of the electron energy
spectrum of bilayer graphene with a strong high-frequency electromagnetic field
(dressing field) results in the Lifshitz transition - the abrupt change in the
topology of the Fermi surface near the band edge. This effect substantially
depends on the polarization of the field: The linearly polarized dressing field
induces the Lifshitz transition from the quadruply-connected Fermi surface to
the doubly-connected one, whereas the circularly polarized field induces the
multicritical point, where the four different Fermi topologies may coexist. As
a consequence, the discussed phenomenon creates physical basis to control the
electronic properties of bilayer graphene with light.
