This is the seventh volume of the collected papers of John Milnor, and much like the other collected volumes [Zbl 0857.01015; Zbl 0857.01016; Zbl 1175.01092; Zbl 1122.01020; Zbl 1187.01036; Zbl 1187.01036; Zbl 1208.01050; Zbl 1278.37006] it is a carefully edited and collated collection that reflects many foundational results from one of the leading mathematicians of the 20th century. In addition to a charming brief dedication to Adrien Douady (1935–2006) including a short account of the atmosphere of excitement, creative energy, and welcome to new participants during the ‘explosion in holomorphic dynamics’ largely driven by Adrien Douady, John Hubbard, Dennis Sullivan, and Bill Thurston, there is a useful introduction to each of the collated papers with an overview of more recent developments and some placing into context by the editor Araceli Bonifant. This, together with some updated references and an index to the whole volume, adds greatly to the utility of the volume for researchers less familiar with the field particularly.
The main emphasis is on Milnor’s contributions to holomorphic dynamics, but there are in addition important papers on real dynamics, an unpublished survey on surjective cellular automata from 1984, an unpublished manuscript from 2000 giving an accessible proof of Tsujii’s result that the topological entropy of \(x\mapsto x^2+c\) is monotonic in the real parameter \(c\), expository lectures on local connectivity of Julia sets, on the moduli space of rational maps with two critical points, on external rays to the Mandelbrot set, on a worked out example of mating, and a detailed exposition of the 1918 paper of Lattès ‘together with its historical antecedents, and its modern formulations and applications’ (an updated form of a paper originally published in 2006). Nine of the papers assembled here have been retypeset into T_EX with some editing for clarity and consistency of notation and updating of references, adding to the clarity and readability of the volume. In addition to the papers and surveys already mentioned, this volume contains: [A. Bonifant et al., “Elliptic curves as attractors in \(\mathbb{P}^2\), Part I: Dynamics”, Exp. Math. 16, No. 4, 385–420 (2007; Zbl 1136.37026)]; [A. Bonifant and the author, “Schwarzian derivatives and cylinder maps”, in: Holomorphic dynamics and renormalization. A volume in honour of John Milnor’s 75th birthday. Proceedings of the workshop on holomorphic dynamics, Toronto, Canada, March 7–11, 2006. Providence, RI: American Mathematical Society (AMS); Toronto: The Fields Institute for Research in Mathematical Sciences. 1–21 (2008; Zbl 1161.37027)]; [the author, “Cubic polynomial maps with periodic critical orbit. I”, in: Complex dynamics. Families and friends. Wellesley, MA: A K Peters. 333–411 (2009; Zbl 1180.37073)]; [A. Bonifant et al., “Cubic polynomial maps with periodic critical orbit. II: Escape regions”, Conform. Geom. Dyn. 14, 68–112 (2010; Zbl 1188.37047)]; [the author, “Hyperbolic components”, Contemp. Math. 573, 183–232 (2012; Zbl 1333.37037)].
This is a remarkable collection of contributions, which has been carefully and thoroughly edited and updated, with much mathematics of great importance. I hope it finds its home not in the dustier parts of libraries with bound volumes of collected works that simply comprise collated papers, but in the shelves and on the desks of active researchers interested in dynamics where it belongs.