Hyperbolic systems with non-computable basins of attraction

In this paper we show that there is an analytic and computable dynamical system with a hyperbolic sink s such that the basin of attraction of s is not computable. As a consequence, the problem of determining basins of attraction for analytic hyperbolic systems is not, in general, algorithmically solvable. 1 Summary of the paper Dynamical systems are fascinating mathematical objects. They can be defined with simple rules and yet their “time-evolution” can be highly complex and difficult to describe analytically. A particular challenge is the problem of characterizing basins of attraction. In recent years, as fast computers have become available, much effort has been devoted to developing algorithms for estimating basins of attraction of various attractors. It therefore becomes useful to know whether or not these sets can actually be generated by computers. It is well known that for hyperbolic rational functions, there are (polynomialtime) algorithms for computing basins of attraction and their complements (Julia sets) with arbitrary precision [Bea91]; in other words, basins of attraction and Julia sets of hyperbolic rational functions are (polynomial-time) computable. However, the question of computability remains open for (analytic) nonrational systems. In this paper we show that: Main Theorem. There is an analytic and computable dynamical system with a hyperbolic sink s such that the basin of attraction of s is not computable (technically more precise results are given in Theorem 5 and Corollary 7 below). Thus our result implies that no algorithmic characterization exists, in general, for a given basin of attraction.