Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions
Abstract
In this note we give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system f:X-->X, including the dynamical degree d(f), which gives a coarse measure of the geometric complexity of the iterates of f, the arithmetic degree a(f,P), which gives a coarse measure of the arithmetic complexity of the orbit of a an algebraic point P in X, and various versions of the canonical height h_f(P) that provide more refined measures of arithmetic complexity. Emphasis is placed on open problems and directions for further exploration.
Cite
@article{arxiv.2408.01559,
title = {Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions},
author = {Joseph H. Silverman},
journal= {arXiv preprint arXiv:2408.01559},
year = {2024}
}
Comments
To appear in the proceedings of the Simons Symposia on Algebraic, Complex, and Arithmetic Dynamics. This article is an expanded version of a talk presented at the Simons Symposium, May 2019, Germany. A small number of updates were added in 2023 and 2024 and are noted as such. 19 pages