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DEGENERATIONS OF COMPLEX DYNAMICAL SYSTEMS

Part of: Arithmetic and non-Archimedean dynamical systems Complex dynamical systems

Published online by Cambridge University Press:  10 April 2014

LAURA DE MARCO  and
XANDER FABER
LAURA DE MARCO
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, USAdemarco@uic.edu
XANDER FABER
Affiliation:
Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI, USAxander@math.hawaii.edu

Abstract

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We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphere ${\hat{{\mathbb{C}}}} $ must be a countable sum of atoms. For a one-parameter family $f_t$ of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit on $\hat{{\mathbb{C}}}$ as the family degenerates. The family $f_t$ may be viewed as a single rational function on the Berkovich projective line $\mathbf{P}^1_{\mathbb{L}}$ over the completion of the field of formal Puiseux series in $t$, and the limiting measure on $\hat{{\mathbb{C}}}$ is the ‘residual measure’ associated with the equilibrium measure on $\mathbf{P}^1_{\mathbb{L}}$. For the proof, we introduce a new technique for quantizing measures on the Berkovich projective line and demonstrate the uniqueness of solutions to a quantized version of the pullback formula for the equilibrium measure on $\mathbf{P}^1_{\mathbb{L}}$.

MSC classification

Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions 37P50: Dynamical systems on Berkovich spaces
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e6
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

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