Superstable Manifolds of Invariant Circles and Co-dimension 1 Böttcher Functions

Let f : X 99K 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 P that is invariant under f , with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose f restricted to this line is given by z 7→ z, with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W 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 loc(S) is not real analytic in the neighborhood of any point.