We define an "ample canonical height" for an endomorphism on a projective
variety, which is essentially a generalization of the canonical heights for
polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical
analogue of the Northcott finiteness theorem for ample canonical heights as a
conjecture, and prove it for endomorphisms on varieties of small Picard
numbers, abelian varieties, and surfaces. As applications, for the
endomorphisms which satisfy the conjecture, we show the non-density of the set
of preperiodic points over a fixed number field, and obtain a dynamical
Mordell--Lang type result on the intersection of two Zariski dense orbits of
two endomorphisms on a common variety.Comment: 41 pages. The previous version has a serious mistake on the proof of
the main conjecture for simple abelian varieties, but the present version
gives a renewed proof that works for arbitrary abelian varietie
