On the number of points with bounded dynamical canonical height

Abstract

This paper discusses the number of points for which the dynamical canonical height is less than or equal to a given value. The height function is a fundamental and important tool in number theory to capture the “number-theoretic complexity” of a point. Asymptotic formulas for the number of points in projective space below a given height have been studied by Schanuel [Sch64], for example, and their coefficients can be written by class numbers, regulators, special values of the Dedekind zeta function, and other number theoretically interesting values. We consider an analogous problem for dynamical canonical height, a dynamical analogue of the height function in number theory, introduced by Call and Silverman [CS93]. The main tool of this study is the dynamical height zeta function studied by Hsia [Hsi97]. In this paper, we give explicit formulas for the dynamical height zeta function in special cases, derive general formulas for obtaining asymptotic behavior from certain functions, and combine them to derive asymptotic behavior for the number of points with bounded dynamical canonical height.