Dynamics of Birational Maps of ℙ2

Inspired by work done for polynomial automorphisms, we apply pluripotential theory to study iteration of birational maps of ℙ2. A major theme is that success of pluripotential theoretic constructions depends on separation between orbits of the forward and backward indeterminacy sets. In particular,...

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Bibliographic Details
Published in:Indiana University mathematics journal 1996-10, Vol.45 (3), p.721-772
Main Author: Diller, Jeffrey
Format: Article
Language:English
Subjects:
Algebra
Automorphisms
Coordinate systems
Critical points
Degrees of polynomials
Integers
Mathematical theorems
Polynomials
Saddle points
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Summary:Inspired by work done for polynomial automorphisms, we apply pluripotential theory to study iteration of birational maps of ℙ2. A major theme is that success of pluripotential theoretic constructions depends on separation between orbits of the forward and backward indeterminacy sets. In particular, we show that a very mild separation hypothesis guarantees the existence of a plurisubharmonic escape function G̃+ and the induced current μ+. We show that under normalized pullback by the birational map, a large class of currents are attracted to μ+. Under stronger separation hypotheses, we establish relationships between the set of normality, stable manifolds of saddle periodic points, and the support of μ+. We illustrate this work in the more concrete setting of quadratic polynomial maps of ℂ2 with merely rational inverses.
ISSN:0022-2518
1943-5258