Superstable manifolds of invariant circles and co-dimension 1 Bottcher functions

We consider the situation of a dominant meromorphic self-map $f: X -rightarrow X$, where $X$ is a compact K\"ahler manifold of dimension $n > 1$. Suppose there is an embedded copy of $\mathbb{P}^1$ 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 $z\mapsto z^b$, with resulting invariant circle $S$. We prove that if $a \geq b$, 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 $a \geq b$ 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.