Polynomial diffeomorphisms of 𝐶². II. Stable manifolds and recurrence

f(x,y)= (y,y 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: