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Eigenvaluations | | Charles Favre
; Mattias Jonsson
; | | Date: |
19 Oct 2004 | | Subject: | Dynamical Systems; Algebraic Geometry MSC-class: 32H50; 14R10, 13A18 | math.DS math.AG | | Abstract: | We study the dynamics in C^2 of superattracting fixed point germs and of polynomial maps near infinity. In both cases we show that the generic asymptotic attraction rate is a quadratic integer, and construct a plurisubharmonic function with the corresponding invariance property. This is done by first finding an infinitely near point at which the map becomes rigid: the critical set is contained in a totally invariant set with simple normal crossings. Our main tool for finding this infinitely near point is to study the action of the dynamics on a space of valuations. This space carries a real tree structure and conveniently encodes local data: an infinitely near point corresponds to a unique open subset of the tree. The induced action respects the tree structure and admits a fixed point--or eigenvaluation--which is attracting in a certain sense. A suitable basin of attraction corresponds to the desired infinitely near point. Our study of polynomial maps also makes essential use of affine curves with one place at infinity. | | Source: | arXiv, math.DS/0410417 | | Services: | Forum | Review | PDF | Favorites | |
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