Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions

TL;DR

This paper reviews key measures of complexity in algebraic dynamical systems, including dynamical degree, arithmetic degree, and canonical height, highlighting open problems and future research directions.

Contribution

It provides a comprehensive overview of complexity measures in algebraic dynamics, emphasizing open problems and potential future research avenues.

Findings

Dynamical degree measures geometric complexity of iterates.

Arithmetic degree assesses arithmetic complexity of orbits.

Canonical heights offer refined arithmetic complexity measures.

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.