On the failure of a generalized Denjoy-Wolff theorem

The classical Denjoy-Wolff Theorem for the unit disk was generalized, in 1988, by Maurice Heins, to domains bounded by finitely many analytic Jordan curves. Heins asked whether such an extension is valid more generally. We show that it can actually fail for some domains. Specifically, we produce an automorphism $\phi$on a planar domain $\Omega$, such that the iterates of $\phi$ converge to a unique Euclidean boundary point, but do not converge to a unique Martin point in the Martin compactification of $\Omega$. We then extend this example to a family of examples in the second part of this work. We thus consider the Martin boundary for domains whose complement is contained in a strip and generalize results of Benedicks and Ancona.