A DYNAMICAL PAIRING BETWEEN TWO RATIONAL MAPS

作者：Petsche Clayton^*; Szpiro Lucien; Tucker Thomas J

来源：Transactions of the American Mathematical Society, 2012, 364(4): 1687-1710.

DOI：10.1090/S0002-9947-2011-05350-X

• Arithmetic dynamical systems
• canonical heights
• equidistribution of small points

Given two rational maps phi and psi on P-1 of degree at least two, we study a symmetric, nonnegative real-valued pairing %26lt;phi, psi %26gt; which is closely related to the canonical height functions h(phi) and h(psi) associated to these maps. Our main results show a strong connection between the value of %26lt;phi, psi %26gt; and the canonical heights of points which are small with respect to at least one of the two maps phi and psi. Several necessary and sufficient conditions are given for the vanishing of %26lt;phi, psi %26gt;. We give an explicit upper bound on the difference between the canonical height h(psi), and the standard height h(st) in terms of %26lt;sigma, psi %26gt;, where sigma(x) = x(2) denotes the squaring map. The pairing %26lt;sigma, psi %26gt; is computed or approximated for several families of rational maps psi.

• 出版日期2012-4