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DEGENERATION OF ENDOMORPHISMS OF THE COMPLEX PROJECTIVE SPACE IN THE HYBRID SPACE

Part of: Classical measure theory Holomorphic mappings and correspondences Non-Archimedean analysis Complex dynamical systems

Published online by Cambridge University Press:  31 August 2018

Charles Favre
Charles Favre*
Affiliation:
CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France (charles.favre@polytechnique.edu)
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Abstract

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.

Keywords

hybrid spacesequilibrium measuresendomorphisms of the complex projective planeholomorphic families of endomorphisms

MSC classification

Primary: 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
Secondary: 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem) 28A33: Spaces of measures, convergence of measures 32H50: Iteration problems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 4 , July 2020 , pp. 1141 - 1183
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Copyright
© Cambridge University Press 2018

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Footnotes

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.

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