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endomorphisms of the complex projective plane

Journal of the Institute of Mathematics of Jussieu (1)

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DEGENERATION OF ENDOMORPHISMS OF THE COMPLEX PROJECTIVE SPACE IN THE HYBRID SPACE
Part of
Charles Favre
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 4 / July 2020

Published online by Cambridge University Press:

31 August 2018, pp. 1141-1183

Print publication:

July 2020
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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.