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Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences

By Eric Bedford and Kyounghee Kim

Abstract

We consider the family $f_{a,b}(x,y)=(y,(y+a)/(x+b))$ of birational maps of the plane and the parameter values $(a,b)$ for which $f_{a,b}$ gives an automorphism of a rational surface. In particular, we find values for which $f_{a,b}$ is an automorphism of positive entropy but no invariant curve. The Main Theorem: If $f_{a,b}$ is an automorphism with an invariant curve and positive entropy, then either (1) $(a,b)$ is real, and the restriction of $f$ to the real points has maximal entropy, or (2) $f_{a,b}$ has a rotation (Siegel) domain.Comment: 24 pages, 7 figures, A companion Mathematica notebook is available at: http://www.math.fsu.edu/~kim

Topics: Mathematics - Dynamical Systems, 37F99, 32M99, 32H50
Year: 2009
OAI identifier: oai:arXiv.org:math/0611297

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