Periodicities in linear fractional recurrences: Degree growth of birational surface maps

By Eric Bedford, Kyounghee Kim. 2005

116 citations recorded by the source corpus.

We consider the set of all 2-step recurrences (difference equations) that are given by linear fractional maps. These give birational maps of the plane. We determine the degree growth of these birational maps. We find the all the maps in this family that are periodic. This also leads to new surface automorphisms with positive entropy.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b986bb9fd4c4ac9725c4e3447dca9df969031ea5

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In Geometry and Topology

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Cremona transformations, surface automorphisms and plane cubics (2008, background): “Bedford and Kim [ BK2 ] found some new examples by studying an explicit family of cremona transformations, i.e.”
On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties (2012, background): “The study of the dynamical degree and its relation to entropy was initiated in [3, 33] and is currently an area of active research; see for example [1, 4, 5, 6, 7, 8, 9, 12, 13, 17, 23, 29, 30, 31, 32, 37].”
Rational Surface Automorphisms with Positive Entropy (2010, background): “Bedford and Kim [3, 4] found some examples of automorphisms by studying an explicit family of quadratic birational maps on P2, and then McMullen [11] gave a synthetic”
Dynamics on blowups of the projective plane (2007, background): “Bedford for bringing [BK2] to my attention.”
Seventy years of Salem numbers (2014, unknown): “Bedford and Kim [6] have shown that this lower bound is realized by a particular rational surface automorphism.”
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 surfaces with a large group of automorphisms (2011, background): “There are classical examples of this kind as well as very recent constructions (see [20], [3], [31], [40]).”
Dynamics of automorphisms of compact complex surfaces (2014, methodology): “We refer to [80, 7, 9, 102] for constructions of examples, to [8, 48] for their deformations, and to [31] for restrictions on the size of such groups of automorphisms.”
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.”