Liang Chung Hsia

Closure of periodic points over a non-archimedean field

The closure of the periodic points of rational maps over a non-archimedean field is studied. An analogue of Montels theorem over non-archimedean fields is first proved. Then, it is shown that the (nonempty) Julia set of a rational map over a non-archimedean field is contained in the closure of the periodic points.

Canonical heights, transfinite diameters, and polynomial dynamics

Abstract Let ϕ(z ) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating ϕ gives rise to a dynamical system and a corresponding canonical height function ĥϕ , as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of ϕ over various completions of K, and we apply this formula to give a generalization of Bilu’s equidistribution theorem for sequences of points whose canonical heights tend to zero.

Preperiodic points for families of rational maps

Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and b are preperiodic for f_l. In particular we show that if P,Q are polynomials over the algebraic numbers such that deg(P) >= 2+deg(Q), and there exists l such that a is periodic for P(x)/Q(x) + l, but b is not preperiodic for P(x)/Q(x) + l, then there exist at most finitely many l such that both a and b are preperiodic for P(x)/Q(x)+l. We also prove a similar result for certain two-dimensional families of endomorphisms of P^2.

On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

AbstractLet K be a function field over finite field

A weak Néron model with applications to p-adic dynamical systems

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On a dynamical Brauer―Manin obstruction

and let

A quantitative estimate for quasiintegral points in orbits

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