Extends Kawaguchi-Silverman conjecture to higher-dimensional subvariety orbits by defining cycle-independent arithmetic degrees and formulating conjectures that relate them to dynamical degrees.
(Relative) dynamical degrees of rational maps over an algebraic closed field
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abstract
The main purpose of this paper is to define dynamical degrees for rational maps over an algebraic closed field of characteristic zero and prove some basic properties (such as log-concavity) and give some applications. We also define relative dynamical degrees and prove a "product formula" for dynamical degrees of semi-conjugate rational maps in the algebraic setting. The main tools are the Chow's moving lemma and a formula for the degree of the cone over a subvariety of $\mathbb{P}^N$. The proofs of these results are valid as long as resolution of singularities are available (or more generally if appropriate birational models of the maps under consideration are available). This observation is applied for the cases of surfaces and threefolds over a field of positive characteristic.
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2019 1verdicts
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Higher arithmetic degrees of dominant rational self-maps
Extends Kawaguchi-Silverman conjecture to higher-dimensional subvariety orbits by defining cycle-independent arithmetic degrees and formulating conjectures that relate them to dynamical degrees.