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

Abstract

We consider the situation of a dominant meromorphic self-map $f:X-rightarrowX$, where $X$ is a compact Kähler 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 $a\ge b$, 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 $a\ge b$ 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.

20 pages, 4 figures, comments welcome, to appear in Ergodic Theory and Dynamical Systems