For surjective endomorphisms of normal projective varieties over characteristic zero fields, the arithmetic degree of any Zariski dense orbit must be a cohomological Lyapunov multiplier; if the first dynamical degree exceeds the second, the Kawaguchi-Silverman conjecture follows.
Numerical action for endomorphisms
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abstract
Let $f: X\to X$ be a surjective endomorphism of a projective variety of dimension $d$. The aim of this paper is to study the action of $f$ on the numerical group of divisors. For exmaple, I proved that $f$ is cohomologically hyperbolic if and only if it is quasi-amplified; and it is amplified if and only if every subsystem of $(X,f)$ is cohomologically hyperbolic. For the proofs, I introduced a notion of spectrum in linear algebra for an open and saliant invariant cone. I also introduce a notion of generated (positive) cycles as an algebraic analogy of (positive) closed current.
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Arithmetic Degrees are Cohomological Lyapunov Multipliers
For surjective endomorphisms of normal projective varieties over characteristic zero fields, the arithmetic degree of any Zariski dense orbit must be a cohomological Lyapunov multiplier; if the first dynamical degree exceeds the second, the Kawaguchi-Silverman conjecture follows.