Eigenvaluations
By C. Favre, Mattias Jonsson. 2004
53 citations recorded by the source corpus.
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 asymptotic attraction rate is a quadratic integer, and construct a plurisubharmonic function with the adequate invariance property. This is done by finding an infinitely near point at which the map becomes rigid: the critical set is contained in a totally invariant set with normal crossings. We locate this infinitely near point through the induced action of the dynamics on a space of valuations. This space carries an real-tree structure and conveniently encodes local data: an infinitely near point corresponds to a open subset of the tree. The 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.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/84c6cc0d1e1d6d6e44c716fbc7fb70bebea60f3f
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In Geometry and Topology
In Mathematical Physics
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Citations in the corpus
- Valuations and plurisubharmonic singularities (2007, background): “It would also be of interest to generalize the dynamical results from [FJ4] to higher dimensions.”
- Degree growth of meromorphic surface maps (2006, methodology): “However, our Main Theorem for polynomial maps does not immediately follow from the analysis in [FJ4]: the methods of the two papers can be viewed as complementary.”