Joseph H. Silverman
Published 2024-08-02Version 1
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.
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
Degeneration of Dynamical Degrees in Families of Maps
Dynamical Degree and Arithmetic Degree of Endomorphisms on Product Varieties
Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space
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