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We introduce a new canonical height function for Jordan blocks of smalleigenvalues for endomorphisms on smooth projective varieties over a numberfield. We prove that under an assumption on the eigenvalues of the endomorphismon the group of divisors modulo numerical equivalence, the arithmetic degree ata rational point is equal to one if and only if it is preperiodic under theendomorphism.

The canonical heights for Jordan blocks of small eigenvalues, preperiodic points, and the arithmetic degrees