  We introduce a notion of stability for equilibrium measures in holomorphic
families of endomorphisms of CP(k) and prove that it is equivalent to the
stability of repelling cycles and equivalent to the existence of some
measurable holomorphic motion of Julia sets which we call equilibrium
lamination. We characterize the corresponding bifurcations by the strict
subharmonicity of the sum of Lyapunov exponents or the instability of critical
dynamics and analyze how repelling cycles may bifurcate. Our methods deeply
exploit the properties of Lyapunov exponents and are based on ergodic theory
and on pluripotential theory.
