POLYNOMIAL DIFFEOMORPHISMS OF d . II : STABLE MANIFOLDS AND RECURRENCE

Friedland and Milnor [FM] have shown that from a dynamical point of view the polynomial diffeomorphisms of C2 fall naturally into two classes. The first class consists of diffeomorphisms with simple dynamics. The diffeomorphisms in this class have periodic points of at most finitely many periods and topological entropy zero. The second class contains the well-known Henon map f(x, y) = (y, i-ax + c). The diffeomorphisms in this class have complicated dynamics: in particular, they have periodic points of infinitely many periods and positive topological entropy (see [FM, S]). We can distinguish between these classes by considering the growth of the degrees of iterates of the diffeomorphism. Define deg(f) to be the maximum of the degrees of the coordinate functions. This quantity is not a conjugacy invariant of f, hence not a dynamical invariant. We can construct a conjugacy invariant, which we call the dynamical degree, as follows: