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

Abstract

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.

32 pages

References in corpus (4)

• Algebraic degrees for iterates of meromorphic self-maps of ${\P }^k$
• Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space
• Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties
• Examples of dynamical degree equals arithmetic degree

Cited by in corpus (4)

• Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space
• Convergence to the Mahler measure and the distribution of periodic points for algebraic Noetherian ${\mathbb{Z}}^d$-actions
• Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties
• Examples of dynamical degree equals arithmetic degree