Hardcover ISBN:  9781470446888 
Product Code:  GSM/198 
463 pp 
List Price:  $95.00 
MAA Member Price:  $85.50 
AMS Member Price:  $76.00 
Electronic ISBN:  9781470451066 
Product Code:  GSM/198.E 
463 pp 
List Price:  $95.00 
MAA Member Price:  $85.50 
AMS Member Price:  $76.00 

Book DetailsGraduate Studies in MathematicsVolume: 198; 2019MSC: Primary 37;
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 nonarchimedean variable is the analogous theory over nonarchimedean fields rather than over the complex numbers. It is also an essential component of the numbertheoretic study of arithmetic dynamics.
This textbook presents the fundamentals of nonarchimedean dynamics, including a unified exposition of RiveraLetelier'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 nonarchimedean analysis. The presentation is accessible to graduate students with only firstyear courses in algebra and analysis under their belts, although some previous exposure to nonarchimedean 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 nonarchimedean dynamics.ReadershipGraduate students and researchers interested in arithmetic and nonarchimedean dynamics.

Table of Contents

Chapters

Introduction

Background

Basic dynamics on $\mathbb {P}^1(K)$

Some background on nonarchimedean fields

Power series and Laurent series

Elementary nonarchimedean dynamics

Fundamentals of nonarchimedean dynamics

Fatou and Julia sets

The Berkovich line

The Berkovich projective line

Rational functions and Berkovich space

Dynamics on the Berkovich line

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 from nonarchimedean analysis

Proofs of results from nonarchimedean analysis

Proofs of Berkovich space results

Proofs of results on Berkovich maps

Appendices

Fatou components without Berkovich space

Other constructions of Berkovich spaces


Additional Material

Reviews

This book would be good for a topics course in Berkovich space for a graduate student familiar with real and complex analysis. [Bennedeto] makes Berkovich space accessible to the new researcher.
Bianca Thompson, MAA Reviews


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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 nonarchimedean variable is the analogous theory over nonarchimedean fields rather than over the complex numbers. It is also an essential component of the numbertheoretic study of arithmetic dynamics.
This textbook presents the fundamentals of nonarchimedean dynamics, including a unified exposition of RiveraLetelier'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 nonarchimedean analysis. The presentation is accessible to graduate students with only firstyear courses in algebra and analysis under their belts, although some previous exposure to nonarchimedean 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 nonarchimedean dynamics.
Graduate students and researchers interested in arithmetic and nonarchimedean dynamics.

Chapters

Introduction

Background

Basic dynamics on $\mathbb {P}^1(K)$

Some background on nonarchimedean fields

Power series and Laurent series

Elementary nonarchimedean dynamics

Fundamentals of nonarchimedean dynamics

Fatou and Julia sets

The Berkovich line

The Berkovich projective line

Rational functions and Berkovich space

Dynamics on the Berkovich line

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 from nonarchimedean analysis

Proofs of results from nonarchimedean analysis

Proofs of Berkovich space results

Proofs of results on Berkovich maps

Appendices

Fatou components without Berkovich space

Other constructions of Berkovich spaces

This book would be good for a topics course in Berkovich space for a graduate student familiar with real and complex analysis. [Bennedeto] makes Berkovich space accessible to the new researcher.
Bianca Thompson, MAA Reviews