In dynamical systems examples are common in which two or more
attractors coexist, and in such cases, the basin boundary is
nonempty. When there 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.
The purpose of this paper is to describe the structure and
properties of basins and their boundaries for two-dimensional
diffeomorphisms. We introduce the basic notion of a ‘basin cell’. A
basin cell is a trapping region generated by some well chosen
periodic orbit and determines the structure of the corresponding
basin. This new notion will play a fundamental 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 $x\in M$ is a Wada point if every open
neighborhood of $x$ has a nonempty intersection with at least
three different basins. We call a basin $B$ a Wada basin if
every $x\in\partial\bar{B}$ 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 least three basins. This result implies conditions
for basins $B_{1},B_{2},\ldots,B_{N}(N\ge 3)$ to satisfy
$\partial\bar{B}_{1}=\partial\bar{B}_{2}=\cdots
=\partial\bar{B}_{N}$.