Superstable manifolds of invariant circles and codimension-one Böttcher functions

Authors

Scott R. Kaschner, Butler UniversityFollow
Roland K.W. Roeder

Document Type

Article

Publication Date

2015

Publication Title

Ergodic Theory and Dynamical Systems

First Page

152

Last Page

175

DOI

http://dx.doi.org/10.1017/etds.2013.39

Abstract

Let f:X ⇢ X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n>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↦zb, with resulting invariant circle S. We prove that if a≥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≥b cannot be relaxed without adding additional hypotheses by presenting two examples with a