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.
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