TY - JOUR

T1 - Higher arithmetic degrees of dominant rational self-maps

AU - Dang, Nguyen Bac

AU - Ghioca, Dragos

AU - Hu, Fei

AU - Lesieutre, John

AU - Satriano, Matthew

N1 - Publisher Copyright:
© 2022 Scuola Normale Superiore. All rights reserved.

PY - 2022

Y1 - 2022

N2 - Suppose that f W X 99K X is a dominant rational self-map of a smooth projective variety defined over Q. Kawaguchi and Silverman conjectured that if P 2 X.Q/ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree !1.f / of f if the orbit of P is Zariski dense in X. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of X. We begin by defining a set of arithmetic degrees of f , independent of the choice of cycles, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.

AB - Suppose that f W X 99K X is a dominant rational self-map of a smooth projective variety defined over Q. Kawaguchi and Silverman conjectured that if P 2 X.Q/ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree !1.f / of f if the orbit of P is Zariski dense in X. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of X. We begin by defining a set of arithmetic degrees of f , independent of the choice of cycles, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.

UR - http://www.scopus.com/inward/record.url?scp=85125162122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85125162122&partnerID=8YFLogxK

U2 - 10.2422/2036-2145.201908_014

DO - 10.2422/2036-2145.201908_014

M3 - Article

AN - SCOPUS:85125162122

SN - 0391-173X

VL - 23

SP - 463

EP - 481

JO - Annali della Scuola normale superiore di Pisa - Classe di scienze

JF - Annali della Scuola normale superiore di Pisa - Classe di scienze

IS - 1

ER -