THE STRUCTURE OF BASINS OF ATTRACTION AND THEIR TRAPPING REGIONS

In dynamical systems examples are common in which two or more attracto rs coexist, and in such cases, the basin boundary is nonempty. When th ere are three basins of attraction, is it possible that every boundary point of one basin is on the boundary of the two remaining basins? Is it possible that all three boundaries of these basins coincide? When this last situation occurs the boundaries have a complicated structure . This phenomenon does occur naturally in simple dynamical systems. Th e purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. We i ntroduce the basic notion of a 'basin cell'. A basin cell is a trappin g region generated by some well chosen periodic orbit and determines t he structure of the corresponding basin. This new notion will play a f undamental role in our main results. We consider diffeomorphisms of a two-dimensional smooth manifold M without boundary, which has at least three basins. A point x is an element of M is a Wada point if every o pen neighborhood of x has a nonempty intersection with at least three different basins. We call a basin B a Wada basin. if every x is an ele ment of partial derivative (B) over bar is a Wada point. Assuming B is the basin of a basin cell (generated by a periodic orbit P), we show that B is a Wada basin if the unstable manifold of P intersects at lea st three basins. This result implies conditions for basins B-1, B-2, . .., B-N(N greater than or equal to 3) to satisfy partial derivative (B ) over bar(1) = partial derivative (B) over bar(2) = ... = partial der ivative (B) over bar(N).