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Superstable manifolds of invariant circles and codimension-one Böttcher functions

Published online by Cambridge University Press:  05 July 2013

SCOTT R. KASCHNER  and
ROLAND K. W. ROEDER
SCOTT R. KASCHNER
Affiliation:
IUPUI Department of Mathematical Sciences, LD Building, Room 270, 402 North Blackford Street, Indianapolis, Indiana 46202-3267, USA email srkaschn@iupui.edurroeder@math.iupui.edu
ROLAND K. W. ROEDER
Affiliation:
IUPUI Department of Mathematical Sciences, LD Building, Room 270, 402 North Blackford Street, Indianapolis, Indiana 46202-3267, USA email srkaschn@iupui.edurroeder@math.iupui.edu
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Abstract

Let $f: X~\dashrightarrow ~X$ 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 ${ \mathbb{P} }^{1} $ 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 {z}^{b} $, with resulting invariant circle $S$. We prove that if $a\geq b$, then the local stable manifold ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ 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 $a\lt b$ for which ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is not real analytic in the neighborhood of any point.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 1 , February 2015 , pp. 152 - 175
Copyright
© Cambridge University Press, 2013 

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