Dynamical Uniform Bounds for Fibers and a Gap Conjecture. (15th November 2019) - Bell, Jason; Ghioca, Dragos; Satriano, Matthew
Abstract: We prove a uniform version of the Dynamical Mordell–Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our 1st result, we assume X is a quasi-projective variety defined over a field K of characteristic 0, endowed with the action of an étale endomorphism Φ , and fXY is a morphism with Y a quasi-projective variety defined over K . Then for any xX(K), if for each yY(K), the set S_x, y:={nNf(Φ ^n(x))=y} is finite, then there exists a positive integer N_x such that S_x, yN_x for each yY(K) . For our 2nd result, we let K be a number field, f:XP^1 is a rational map, and Φ is an arbitrary endomorphism of X . If O_Φ (x) denotes the forward orbit of x under the action of Φ , then either f(O_Φ (x)) is finite, or _n→ ∞ h(f(Φ ^n(x)))/(n)>0, where h(· ) represents the usual logarithmic Weil height for algebraic points.