Unstable periodic orbits and the symbolic dynamics of the complex Hénon map

A numerical technique for the calculation of unstable periodic orbits of chaotic maps in the complex plane is presented. Applying this technique to the complex Hénon map we show that it can find all the $2^{\mathit{p}}$ real and complex periodic orbits of any given order p. The real periodic orbits coincide with those obtained by a similar algorithm for the ordinary Hénon map that we have proposed earlier, and thus verify its completeness. The method provides a new definition of symbolic dynamics for the Hénon map, which holds for both the real and complex periodic orbits. It provides a computational framework that applies to all limits of this map on a common footing, unlike the conventional techniques in which one needs different algorithms for calculations involving strange attractors, strange repellers, and various types of Julia sets.