Superstable Manifolds of Invariant Circles

Let f : X → X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n > 1. Suppose there is an embedded copy of P1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose also that f restricted to this line is given by z → z ^b , with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W ^s _loc (S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a ≥ b cannot be relaxed without adding additional hypotheses by presenting two examples with a < b for which W ^s _loc (S) is not real analytic in the neighborhood of any point.