For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, we prove that there exists a Zariski dense set of rational points having disjoint orbits if the endomorphism is an automorphism.
Let X be a smooth projective variety defined over Q, and f : X X be a dominant rational map. Let δ f be the first dynamical degree of f and h X : X(Q) −→ [1, ∞) be a Weil height function on X associated with an ample divisor on X. We prove several inequalities which give upper bounds of the sequence (h X (f n (P ))) n≥0 where P is a point of X(Q) whose forward orbit by f is well-defined. As a corollary, we prove that the upper arithmetic degree is less than or equal to the first dynamical degree; α f (P ) ≤ δ f . Furthermore, we prove the canonical height functions of rational self-maps exist under certain conditions. For example, when the Picard number of X is one, f is algebraically stable and δ f > 1, the limit defining canonical height limn→∞ h X (f n (P )) δ n f converges.
In this paper, we consider the limitwhere f : X ÝÑ X is a surjective self-morphism on a smooth projective variety X over a number field, S is a finite set of places, λ Y,v is a local height function associated with a proper closed subscheme Y Ă X, and h H is an ample height function on X. We give a geometric condition which ensures that the limit is zero, unconditionally when dim Y " 0 and assuming Vojta's conjecture when dim Y ě 1. In particular, we prove (one is unconditional, one is assuming Vojta's conjecture) Dynamical Lang-Siegel type theorems, that is, the relative sizes of coordinates of orbits on P N are asymptotically the same with trivial exceptions. These results are higher dimensional generalization of Silverman's classical result.
We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group endomorphisms of linear algebraic groups. We also propose a possible approach to the conjecture using equivariant MMP.Cartier divisors D). By definition, N 1 (X) and N 1 (X) are dual to each other.-When X is normal, the Iitaka dimension of a Q-Cartier divisor D on X is denoted by κ(D). • Let M be a Z-module. We write M Q = M⊗ Z Q, M R = M⊗ Z R, and so on.
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