This became the basis for his first well-known published result known as the "Sternberg linearization theorem" which asserts that a smooth map near a hyperbolic fixed point can be made linear by a smooth change of coordinates provided that certain non-resonance conditions are satisfied. Also proved were generalizations of the Birkhoff canonical form theorems for volume preserving mappings in n-dimensions and symplectic mappings, all in the smooth case. Also, in a sequel to this paper written jointly with Victor Guillemin and Daniel Quillen , he extended this classification to a larger class of pseudogroups : the primitive infinite pseudogroups. One important by-product of the GQS paper was the " integrability of characteristics" theorem for over-determined systems of partial differential equations. This figures in GQS as an analytical detail in their classification proof but is nowadays the most cited result of the paper. Among his contributions to this subject are his paper with Bertram Kostant on BRS cohomology, his paper with David Kazhdan and Bertram Kostant on dynamical systems of Calogero type and his paper with Victor Guillemin on the "Quantization commutes with reduction" conjecture.
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Start your review of Dynamical Systems Write a review Shelves: science , reviewed This has got the be the messiest book I have ever read, math or non-math. The number of typos is unbelievable. At some points whole paragraphs were missing, at other, some paragraphs apparently were copied-and-pasted twice, and then some LaTeX commands pop up in the middle of a sentence. Why is one interested in fixed point theorems? A famous example is the Newton iteration, and this is in fact the topic of the first chapter of this book.
This chapter, together with chapter 8, is already the most difficult one, so that the rest of the book is not too hard to follow. The difficulty ranges from elementary calculus to serious real analysis, so it is manageable. What I particularly liked about the book is that it uses and encourages an experimental use of mathematics, that is, doing numerical experiments, plotting graphs of functions to find fixed points or periodic points and then, after the experiment, supply a proof to confirm the observations.
The book is very efficient in the sense that it progresses to the main results without much ado. Most of the proofs are easy to follow, though the aforementioned typos and some random changes in notation can lead to confusion. From chapter 9 on, the chapters seem hastily slammed together, there is much less cohesion than in the first part of the book, and the motivation for what is done is much less clear.
Based on the first eight chapters, I would have given the book four stars, but as a whole, I cannot bring myself to award more than three.
Dynamical Systems by Shlomo Sternberg