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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Lower bounds for non-Archimedean Lyapunov exponents


Author: Kenneth Jacobs
Journal: Trans. Amer. Math. Soc. 371 (2019), 6025-6046
MSC (2010): Primary 37P50, 11S82; Secondary 37P05
DOI: https://doi.org/10.1090/tran/7344
Published electronically: February 1, 2019
MathSciNet review: 3937317
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Abstract: Let $ K$ be a complete, algebraically closed, non-Archimedean valued field, and let $ \mathrm {\bold P}^1$ denote the Berkovich projective line over $ K$. The Lyapunov exponent for a rational map $ \phi \in K(z)$ of degree $ d\geq 2$ measures the exponential rate of growth along a typical orbit of $ \phi $. When $ \phi $ is defined over $ \mathbb{C}$, the Lyapunov exponent is bounded below by $ \frac {1}{2}\log d$. In this article, we give a lower bound for $ L(\phi )$ for maps $ \phi $ defined over non-Archimedean fields $ K$. The bound depends only on the degree $ d$ and the Lipschitz constant of $ \phi $. For maps $ \phi $ whose Julia sets satisfy a certain boundedness condition, we are able to remove the dependence on the Lipschitz constant.


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Additional Information

Kenneth Jacobs
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: kjacobs@math.northwestern.edu

DOI: https://doi.org/10.1090/tran/7344
Keywords: Lyapunov exponent, non-Archimedean, rational map, lower bound, distortion
Received by editor(s): October 18, 2016
Received by editor(s) in revised form: March 6, 2017, and May 10, 2017
Published electronically: February 1, 2019
Additional Notes: The author gratefully acknowledges support from NSF grant DMS-1344994 of the RTG in Algebra, Algebraic Geometry, and Number Theory, at the University of Georgia.
Article copyright: © Copyright 2019 American Mathematical Society