On the degree growth of birational mappings in higher dimension

By Eric Bedford, Kyounghee Kim. 2004

68 citations recorded by the source corpus.

Let ƒ be a birational map of Cd,and consider the degree complexity or asymptotic degree growth rate δ(ƒ) = limn → ∞ (deg(ƒn))1/n.We introduce a family of elementary maps, which have the form ƒ = L o J, where L is (invertible) linear, and J(x1−1,..., xd) = (x1−1,...,xd−1.We develop a method of regularization and show how it can be used to compute δ for an elementary map.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6aa9a899e004b841e4a3243aa8285edcaeba1d01

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Periodicities in linear fractional recurrences: Degree growth of birational surface maps (2005, background): “ase analysis. We avoid this by adopting a more general approach. In §3 we show how δ(fα,β) may be derived from the set of numbers in open and closed orbit lists. We do this by showing that results of [BK] may be extended from the “elementary” case to the general case. We use this in §4 to determine ”
Cremona transformations, surface automorphisms and plane cubics (2008, background): “When the nj are large enough (see [ BK1 , Theorem 5.1]), e.g.”
Degree-growth of monomial maps (2006, background): “More recent articles on the topic of dynamical degree, intersections, and algebraic entropy include [ BK ], [BFJ], [DF], [DS], [FJ], [T1], [T2], and [TEGORS].”
Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space (2011, background): “See [4, 8, 9] for some discussion of these issues and for the computation of δφ for certain higher-dimensional maps.”
The diminished base locus is not always closed (2012, methodology): “This was studied in dimension 2 in the work of McMullen [13] and Bedford-Kim [2],[3], and in higher dimension by PerroniZhang [17].”
Degree growth of meromorphic surface maps (2006, background): “Even in simple families of mappings, degree growth exhibits a rich behavior: see e.g. the papers by Bedford and Kim [BK1, BK2], which also contain references to the physics literature.”
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].”