Published online by Cambridge University Press: 31 August 2018
We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk. We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holds in the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.
The author is supported by the ERC-starting grant project ‘Nonarcomp’ no. 307856, and by the Brazilian project ‘Ciência sem fronteiras’ founded by the CNPq.