Examples of dynamical degree equals arithmetic degree

By Shu Kawaguchi, Joseph H. Silverman. 2012

47 citations recorded by the source corpus.

Let f : X --> X be a dominant rational map of a projective variety defined over a number field. An important geometric-dynamical invariant of f is its (first) dynamical degree d_f= lim SpecRadius((f^n)^*)^{1/n}. For algebraic points P of X whose forward orbits are well-defined, there is an analogous (upper) arithmetic degree a_f(P) = limsup h_X(f^n(P))^{1/n}, where h_X is an ample Weil height on X. In an earlier paper, we proved the fundamental inequality a_f(P) \le d_f and conjectured that a_f(P) = d_f whenever the orbit of P is Zariski dense. In this paper we show that the conjecture is true for several types of maps. In other cases, we provide support for the conjecture by proving that there is a Zariski dense set of points with disjoint orbits and satisfying a_f(P) = d_f.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/77fb53b162877e7145604b7ee6f19d2749200eba

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Current trends and open problems in arithmetic dynamics (2018, background): “, fN ] such that their coordinate functions fi are monomials [215], and parts of the conjecture are known in various other cases; see for example [128, 133, 134, 135].”
On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties (2012, background): “The proofs of these results, together with other cases for which we can prove the weaker statement that αf(P ) = δf(X) for a Zariski dense set of points P ∈ Xf(K̄) having disjoint orbits, will appear in a companion publication [25].”