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Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences

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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.

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Authors and Affiliations

  1. Indiana University, Bloomington, IN, 47405, USA

    Eric Bedford

  2. Florida State University, Tallahassee, FL, 32306, USA

    Kyounghee Kim

Authors
  1. Eric Bedford
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  2. Kyounghee Kim
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Correspondence to Kyounghee Kim.

Additional information

Communicated by John Erik Fornaess.

Research supported in part by the NSF.

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Bedford, E., Kim, K. Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences. J Geom Anal 19, 553–583 (2009). https://doi.org/10.1007/s12220-009-9077-8

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  • DOI: https://doi.org/10.1007/s12220-009-9077-8

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