Dynamics in one non-archimedean variable
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Bibliographic Information
Dynamics in one non-archimedean variable
（Graduate studies in mathematics, v. 198）
American Mathematical Society, c2019
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
Includes bibliographical references (p. 449-456) and index
Description and Table of Contents
Description
The theory of complex dynamics in one variable, initiated by Fatou and Julia in the early twentieth century, concerns the iteration of a rational function acting on the Riemann sphere. Building on foundational investigations of $p$-adic dynamics in the late twentieth century, dynamics in one non-archimedean variable is the analogous theory over non-archimedean fields rather than over the complex numbers. It is also an essential component of the number-theoretic study of arithmetic dynamics.
This textbook presents the fundamentals of non-archimedean dynamics, including a unified exposition of Rivera-Letelier's classification theorem, as well as results on wandering domains, repelling periodic points, and equilibrium measures. The Berkovich projective line, which is the appropriate setting for the associated Fatou and Julia sets, is developed from the ground up, as are relevant results in non-archimedean analysis. The presentation is accessible to graduate students with only first-year courses in algebra and analysis under their belts, although some previous exposure to non-archimedean fields, such as the $p$-adic numbers, is recommended. The book should also be a useful reference for more advanced students and researchers in arithmetic and non-archimedean dynamics.
Table of Contents
Introduction
Basic dynamics on $\mathbb{P}^1(K)$
Some background on non-archimedean fields
Power series and Laurent series
Fundamentals of non-archimedean dynamics
Fatou and Julia sets
The Berkovich projective line
Rational functions and Berkovich space
Introduction to dynamics on Berkovich space
Classifying Berkovich Fatou components
Further results on periodic components
Wandering domains
Repelling points in Berkovich space
The equilibrium measure
Proofs of results from non-archimedean analysis
Proofs of Berkovich space results
Proofs of results on Berkovich maps
Fatou components without Berkovich space
Other constructions of Berkovich spaces
Bibliography
Index.
by "Nielsen BookData"
