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Böttcher coordinates at fixed indeterminacy points

Published online by Cambridge University Press:  13 March 2018

KOHEI UENO
KOHEI UENO*
Affiliation:
Daido University, Nagoya 457-8530, Japan email k-ueno@daido-it.ac.jp
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Abstract

We first consider the dynamics of a class of meromorphic skew products having superattracting fixed points or fixed indeterminacy points at the origin. Our theorem asserts that, if a map has a suitable weight, then it is conjugate to the associated monomial map on an invariant open set whose closure contains the origin. We next extend this result to a wider class of meromorphic maps such that the eigenvalues of the associated matrices are real and greater than $1$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 12 , December 2019 , pp. 3437 - 3456
Copyright
© Cambridge University Press, 2018 

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References

Böttcher, L. E.. The principal laws of convergence of iterates and their application to analysis. Izv. Kazan. Fiz.-Mat. Obshch. 14 (1904), 137–152 (in Russian).Google Scholar
Buff, X., Epstein, A. L. and Koch, S.. Böttcher coordinates. Indiana Univ. Math. J. 61 (2012), 1765–1799.Google Scholar
Favre, C.. Classification of 2-dimensional contracting rigid germs and Kato surfaces: I. J. Math. Pures Appl. (9) 79 (2000), 475–514.Google Scholar
Favre, C. and Jonsson, M.. Eigenvaluations. Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), 309–349.Google Scholar
Hubbard, J. H. and Papadopol, P.. Superattractive fixed point in C n . Indiana Univ. Math. J. 43 (1994), 321–365.Google Scholar
Milnor, J.. Dynamics in One Complex Variable (Annals of Mathematics Studies, 160) . Princeton University Press, Princeton, NJ, 2006.Google Scholar
Shinohara, T.. Another construction of a Cantor bouquet at a fixed indeterminate point. Kyoto J. Math. 50 (2010), 205–224.Google Scholar
Ueda, T.. Complex Dynamical Systems on Projective Spaces (Advanced Series in Dynamical Systems, 13) . World Scientific, Singapore, 1993, pp. 120–138.Google Scholar
Ueno, K.. Böttcher coordinates for polynomial skew products. Ergod. Th. & Dynam. Sys. 36 (2016), 1260–1277.Google Scholar
Ueno, K.. Böttcher coordinates at superattracting fixed points of holomorphic skew products. Conform. Geom. Dyn. 20 (2016), 43–57.Google Scholar
Ushiki, S.. Böttcher’s Theorem and Super-stable Manifolds for Multidimensional Complex Dynamical Systems (Advanced Series in Dynamical Systems, 11) . World Scientific, Singapore, 1992, pp. 168–184.Google Scholar