Height Bound and Preperiodic Points for Jointly Regular Families of Rational Maps

Lee, Chong Gyu

doi:10.4134/jkms.2011.48.6.1171

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2022

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“…This conjecture was verified for affine plane automorphisms by Denis [Den95], for regular automorphisms and for triangular automorphisms by Marcello [Mar00,Mar03]. See also [Lee11,Lee18,Sil06] for some variants.…”

Section: Introductionmentioning

confidence: 89%

Periodic points and arithmetic degrees of certain rational self-maps

Wang¹

2022

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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 affine automorphisms. We also study the Kawaguchi-Silverman conjecture concerning the dynamical and the arithmetic degrees for birational 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.

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