Superstable manifolds of invariant circles and codimension-one Böttcher functions

<p> Let f:X ⇢ X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n&gt;1. Suppose that 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 that f restricted to this line is given by z&mapsto;zb, with resulting invariant circle S. We prove that if a&ge;b, then the local stable manifold Wsloc(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&ge;b cannot be relaxed without adding additional hypotheses by presenting two examples with a <b> </b></p>