For a dominant rational self-map on a projective variety, the height associated with a closed subscheme vanishes relative to an ample height along generic orbits whenever a dynamical degree is strictly smaller than the arithmetic degree.
Dynamical Degrees, Arithmetic Degrees, and Canonical Heights: History, Conjectures, and Future Directions
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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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Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties
For a dominant rational self-map on a projective variety, the height associated with a closed subscheme vanishes relative to an ample height along generic orbits whenever a dynamical degree is strictly smaller than the arithmetic degree.