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arXiv · 2201.12750

Periodic points and arithmetic degrees of certain rational self-maps

Long Wang
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Abstract

Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the second dynamical degrees distinct. We give a boundedness result about heights of its periodic points. This is motivated by a conjecture of Silverman for polynomial automorphisms of affine spaces. We also study the Kawaguchi--Silverman conjecture concerning dynamical and arithmetic degrees for certain rational self-maps in dimension two. In particular, we reduce the problem to the dynamical Mordell--Lang conjecture and verify the Kawaguchi--Silverman conjecture for some new cases. As a byproduct of the argument, we show the existence of Zariski dense orbits in these cases.

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Long Wang. 2022-01-30. Periodic points and arithmetic degrees of certain rational self-maps. https://arxiv.org/abs/2201.12750

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