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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. 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W. Roeder, Scott R. Kaschner","submitted_at":"2012-08-15T01:51:13Z","abstract_excerpt":"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. 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