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Dynamics in One Non-Archimedean Variable

Robert L. Benedetto
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics (Volume 198)
[Reviewed by
Bianca Thompson
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When studying non-archimedean spaces, like \(\mathbb{Q}_p\), many facts from complex and real analysis still hold. In fact, because of the strong triangle inequality, many of these facts become more straightforward to prove. This makes spaces like \(\mathbb{Q}_p\) a delight to study, that is until you care about connectedness. In steps the theory of Berkovich space to fill in the gap. Vladimir Berkovich introduced Berkovich spaces in the late 80s and it turned out to be a good space to work on non-archimedean dynamics. Now we had a whole new world of dynamical questions to ask. What would it mean to iterate a rational function on Berkovich space? What do the Julia and Fatou sets look like here? In fact, how should we define the Julia and Fatou sets? This book delves into these topics, splitting the book into 5 main parts. 
Part 1 of the book read as a quick refresher on how dynamics works over \(\mathbb{Q}\) and \(\mathbb{C}\) and non-archimedean fields. Part 2 delved into how Fatou sets and Julia sets work in a non-archimedean setting. Parts 3 and 4 focused on what I had wanted out of the book: creating Berkovich space and the different dynamics on this space. Here is where you’ll find the definitions of the Berkovich Fatou and Julia sets. I like how the author chose to place many of the proofs of properties of Berkovich space into part 5 of the book, so as not to slow the pace down for those interested in the dynamics.  The author’s introduction suggests several possible paths through the book depending on your interests, including some very non-linear routes if you only care about Berkovich space and little about dynamics. Since I was interested in the dynamics of Berkovich space and was unfamiliar with much of the non-archimedean background, I originally opted for his suggestion “read the whole book.” Turns out that way is a little daunting, and I found searching through the book to learn specific topics and flipping back as needed to be a better way for me to use the book. 
This book pulls together many fields including dynamics, real, complex and p-adic analysis, and topology all into one place- the Berkovich projective line. It is full of many unsettling facts for someone who has been teaching real analysis for a few years, such as the fact that you can have a decreasing sequence of closed disks with an empty intersection. This is where Type IV points come from in Berkovich space. Benedetto’s proofs are thorough; at the same time, he likes to leave things for the reader so they can better engage with the book. His exercises in chapter 6 about seminorm properties can really help the reader review analysis and understand what the points on Berkovich space really are. 
Each time Benedetto introduced a concept he tried to tie it to a concrete example. This was especially useful when trying to understand how to construct the Berkovich points. For example in Chapter 6, he introduced how to visualize and move between the different points and linked it back to the concept that the points are associated to disks. 
This book would be good for a topics course in Berkovich space for a graduate student familiar with real and complex analysis. Benedetto takes the time to build up and give many examples of each related topic in each chapter. His book was easy to navigate as I began to skip around the chapters. The list of all the notation in the first pages was incredibly helpful, as these objects often had complex definitions that were hard to recall when first learning them, like the definition of local degree. 
Benedetto does a good job of pulling together the many sources that had originally written on Berkovich space: Vladmir Berkovich's original construction, Rivera-Letelier's many results, Potential Theory on the Berkovich Projective Line by Baker and Rumely, and many others. He puts everything into a common language in the book, and, with almost every chapter having a section on examples, he makes Berkovich space accessible to the new researcher.


Bianca Thompson is an assistant professor at Westminster College Salt Lake City, UT. Her research area is number theory and arithmetic dynamics and she has a passion for creating an inclusive environment in her classroom. Her website is at and email address is bthompson[at]westminstercollege[dot]edu.

See the publisher's web page.