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.
Recent advances on Kawaguchi-Silverman conjecture
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This is a survey on Kawaguchi-Silverman conjecture.
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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.