Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences

Details Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences

2006-11-09

arxiv.org/abs/math/0611297v4

Mathematics

Dynamical Systems

24 pages, 7 figures, A companion Mathematica notebook is available at: http://www.math.fsu.edu/~kim/

Scientific paper

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.

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