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arXiv · 2408.01559

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

Joseph H. Silverman
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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.

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Joseph H. Silverman. 2024-08-02. Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions. https://arxiv.org/abs/2408.01559

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