Basins of Wada*

We are accustomed to the idea that while the boundary between two regions is likely to consist of arcs, the points that are in the boundary of three or more regions are intuitively expected to be isolated. In the United States, there is only one point that is in the boundary of four states. In dynamical systems, the boundaries are more complicated. Topologists discovered that it is possible to have three or more regions B i, each an open set such that if p is in the boundary of any one Bi, then it is in fact in the boundary of all the se t s B i. Such a possibility is hard to imagine. We will say that the sets B~ satisfy the Wada property if each boundary point of any B~ is in the boundary of all the B~. The first explicit example [22] was attributed to Yoneyama. See also ref. [15]. We begin in section 2 by describing the example called "The Lakes of Wada". The main point of this paper is that the Wada property may be quite common in dynamics. In section 3 we give an example using the Poincar6 return map of the forced damped pendulum, choosing parameters