Canonical Heights, Transfinite Diameters, and Polynomial Dynamics

Transfinite diameter and the resultant

We prove a formula for the Fekete-Leja transfinite diameter of the pullback of a set E in C^N by a regular polynomial map F, expressing it in terms of the resultant of the leading part of F and the transfinite diameter of E. We also establish the nonarchimedean analogue of this formula. A key step in the proof is a formula for the transfinite diameter of the filled Julia set of F.

Mellin变换、超越直径与积分的有理逼近

Mellin transforms, transfinite diameter and rational approximations of integrals

本文提出一个高维无理性准则，用于证明由Mellin积分表示的周期的无理性，通过超越直径上界分析多变量Vandermonde矩阵。

Consequences of the Continuity of the Monic Integer Transfinite Diameter

We consider the problem of determining the monic integer transfinite diameter for real intervals $I$ of length less than 4. We show that $t_M([0,x])$, as a function in $x>0$, is continuous, therefore disproving two conjectures due to Hare and Smyth. Consequently, for $n>2\in\naturals$, we define the quantity $b_{\max}(n)&=&\sup_{b>\frac{1}{n}}\left\{b|t_M([0,b])=\tfrac{1}{n}\right.\right\}$ and give lower and upper bounds of $b_{\max}(n)$. Finally, we improve the lower bound for $b_{\max}(n)$ for $3\leq n\leq 8$.

The monic integer transfinite diameter

We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I. The monic integer transfinite diameter t_M(I) is defined as the infimum of all such supremums. We show that if I has length 1 then t_M(I) = 1/2. We make three general conjectures relating to the value of t_M(I) for intervals I of length less that 4. We also conjecture a value for t_M([0, b]) where 0 < b < 1. We give some partial results, as well as computational evidence, to support these conjectures. We define two functions that measure properties of the len…

动态典范高度与有限树

Dynamical Canonical Heights and Finite Trees

本文针对素数幂次的特定多项式，基于Dimitrov的思路，通过构建有限树并结合核心熵分析，建立了游荡点典范高度的平方级衰减下界。

Equidistribution of small points, rational dynamics, and potential theory

If phi(z) is a rational function on P^1 of degree at least 2 with coefficients in a number field k, we compute the homogeneous transfinite diameter of the v-adic filled Julia sets of phi for all places v of k by introducing a new quantity called the homogeneous sectional capacity. In particular, we show that the product over all places of these homogeneous transfinite diameters is 1. We apply this product formula and some new potential-theoretic results concerning Green's functions on Riemann surfaces and Berkovich spaces to prove an adelic equidistribution theorem for dynamical systems on the…