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

TL;DR

This paper investigates the regularity of local stable manifolds of invariant circles in meromorphic self-maps on compact Kähler manifolds, establishing conditions for real analyticity based on the map's local dynamics.

Contribution

It proves that the local stable manifold is real analytic when the map's degree exceeds or equals the transverse degree, and provides counterexamples showing this condition is necessary.

Findings

Stable manifold is real analytic if $a \,\geq\, b$.

Counterexamples show analyticity fails if $a < b$.

Localized version of the theorem extends applicability.

Abstract

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.