Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences

By Eric Bedford, Kyounghee Kim. 2006

106 citations recorded by the source corpus.

We consider the family fa,b(x,y)=(y,(y+a)/(x+b)) of birational maps of the plane and the parameter values (a,b) for which fa,b gives an automorphism of a rational surface. In particular, we find values for which fa,b is an automorphism of positive entropy but no invariant curve. The Main Theorem: If fa,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) fa,b has a rotation (Siegel) domain.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7dea4548639d6b7e88297283731d2ab6f22fc199

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Explicit Examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy (2013, background): “or the study in this paper. 1.1. GeneralProblem. Complex dynamics of biholomorphic automorphisms of compact Ka¨hler surfaces is now fairly well-understood since Cantat [Ca99]. Especially, BedfordKim ([BK09], [BK12], [BK10]) and McMullen ([Mc02], [Mc07], [Mc11], [Mc13]) show very beautiful aspects of”
Dynamics on blowups of the projective plane (2007, background): “1 A negative answer in the minimal case is announced in [BK1].”
Degree growth of birational maps of the plane (2011, unknown): “In [2, 3, 4, 5, 12], automorphisms with positive entropy are constructed sta rting from a birational map ofP2.”
Continuous families of rational surface automorphisms with positive entropy (2008, background): “Some of the maps defined in [1] and [2] have invariant curves, but it seems that most of them do not.”
Dynamical degrees of birational transformations of projective surfaces (2013, background): “If char(k) = 0, there is an automorphism g of a rational surface X such that λ(g) = λL (see [4,5] and [37]); McMullen recently announced that such an example also exists on a projective K3 surface (see [38]).”
Rational Surface Automorphisms with Positive Entropy (2010, background): “nal surfaces, rather few examples had been known (see [5], Section 2). However, some rational surface automorphisms with invariant anticanonical curves have been constructed recently. Bedford and Kim [3, 4] found some examples of automorphisms by studying an explicit family of quadratic birational m”
Reflection groups in algebraic geometry (2006, background): “out the minimality condition the necessity of this condition was conjectured by M. Gizatullin but a counter-example was found by B. Harbourne [54]. A recent preprint of Eric Bedford and Kyounghee Kim [8] contains an example of a minimal surface with inﬁnite automorphism group with |−mKX|= ∅for all m”