Computability of julia sets by Mark Braverman and Michael Yampolsky, Publisher: Springer, 2009

The study of dynamical systems has at its core, as has often been noted, a very computational feel. In complex dynamics, one considers the one point compactification Ĉ of the compact plane (one can think of the complex plane itself without too much loss; one can also think of Riemann manifolds more generally), and a rational map f : Ĉ→ Ĉ, and studies the sequence F = {f, f ◦f, f ◦f ◦f, . . . }. We will write F(x) for the sequence of points {f(x), f ◦ f(x), f ◦ f ◦ f(x) . . . }, technically called the orbit of x under f . The inherently recursive nature of the subject, as well as the distinguished history of computer simulation in this area connects the subject closely with computation. Some of the most recognizable icons of the subject are the Julia sets, the enigmatic, often paisley-like images produced by identifying regions in which F exhibits sensitive dependence on initial conditions — that is, small differences in x lead to large differences in the behavior of the sequence F(x). We denote the Julia set for f by Jf . The complement of the Julia set is called the Fatou set. A natural question is whether there is an algorithm to determine whether a given point is in the Julia set or the Fatou set. Obviously, such a determination would require full-precision complex arithmetic, and so is not literally possible. The book under review approaches this question by treating it approximately: Given f , is there a Turing machine M such that, for any n, if we give M a program which computes the digits of the coordinates of a point, then M will output 1 if the point is within 2−n of the Julia set, 0 if it is at least 2−n+1 away from the Julia set, and either 1 or 0 in every other case? In effect, is there an algorithm whose input is a degree of precision and whose output is a plot (of appropriate granularity), coloring pixels black or white, as appropriate? (This approach differs markedly from the earlier approach of Blum, Shub, and Smale, which calls nearly all Julia sets non-computable, but is attractive from different perspectives; see [2]) Thus, a slightly informal restatement of the central problem of the book is this: Question: Given a rational self-map of Ĉ, along with a point p and a radius r, can one determine whether the dynamics of the map become chaotic within r of p?