On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties

By Shu Kawaguchi, Joseph H. Silverman. 2012

91 citations recorded by the source corpus.

Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) << (d_f + e)^n h_X(P), where the implied constant depends only on X, h_X, f, and e. As applications, we prove a fundamental inequality a_f(P) \le d_f for the upper arithmetic degree and we construct canonical heights for (nef) divisors. We conjecture that a_f(P) = d_f whenever the orbit of P is Zariski dense, and we describe some cases for which we can prove our conjecture.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fe271423166952bde8315ac4ddf8f100c51e6d0c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Shu Kawaguchi
2. Joseph H. Silverman
3. Franccois Berteloot
4. Rafe Jones

In Statistical and Nonlinear Physics

1. Joseph H. Silverman
2. Shu Kawaguchi

Citations in the corpus

• Current trends and open problems in arithmetic dynamics (2018, unknown): “3 (Arithmetic Degree Conjectures (on P ): Kawaguchi–Silverman, 2006, [134]; see also [136, 137]).”
• Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space (2011, methodology): “We mention in particular a paper of Jonsson and Wulcan [37] in which they prove much of Conjectures 1 and 3 for polynomial morphisms φ : A(2) → A(2) of small topological degree, and a paper of Kawaguchi and the author [40] in which it is shown that αφ(P ) ≤ δφ holds for dominant rational self-maps o”
• Addendum to “Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties" (2013, background): “. Further, the collection of arithmetic degrees αf(x) : x ∈ X(K¯) is a finite set of algebraic integers. Theroem 1 proves, for morphisms, various conjectures stated more generally for rational maps in [5, 9]. The proof of Theroem 1 uses an extension of the classical theory of dynamical canonical heig”
• Examples of dynamical degree equals arithmetic degree (2012, background): “A principal result of [17] is the fundamental inequality αf (P ) ≤ δf for all P ∈ X(Q)f .”