Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of CP(k)

François Berteloot, Fabrizio Bianchi, Christophe Dupont

Published 2014-03-29, updated 2015-04-23Version 2

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 a measurable holomorphic motion of Julia sets. 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.