1. Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions
- Author
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Silverman, Joseph H.
- Subjects
Mathematics - Number Theory ,Mathematics - Dynamical Systems ,Primary: 37P05, Secondary: 37P15, 37P30, 37P55 - 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., Comment: 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
- Published
- 2024