AI 中文总结
该研究针对非阿基米德完备域上的Berkovich射影直线，计数不含吸引点的非吸引子树，证明其次数和等于有理函数次数，并得到其测度等分布于平衡测度的结果，解答了相关问题。
AI 中文摘要
考虑有理函数在关于非平凡非阿基米德绝对值完备的代数闭域上的Berkovich射影直线上的作用，若不动点轨迹的连通分支不含吸引点，则称其为“非吸引子树”。我们根据有理函数在非吸引子树各点处的局部次数定义“非吸引子树的次数”，证明所有非吸引子树的次数之和等于有理函数的次数。该计数结果通过研究离散测度的势得到，该离散测度以每个非吸引子树的次数为质量。我们证明，当有理函数迭代时，这些测度会等分布到平衡测度；在特定条件下（尤其多项式情形），这简化为排斥点的等分布。我们还证明，当李雅普诺夫指数为正时，II型排斥点在迭代中变得可忽略。这些结果对Favre和Rivera-Letelier的问题给出了部分和完全解答。
英文摘要
Consider the action of a rational function on the Berkovich projective line over an algebraically closed field complete with respect to a nontrivial non-archimedean absolute value. We say a connected component of the fixed locus is a "non-attracting subtree" if it contains no attracting points. We define a "degree of a non-attracting subtree," in terms of the local degrees of the rational function at its points. We show that the degrees of non-attracting subtrees sum up to the degree of the rational function. This counting result is obtained by studying the potential of a discrete measure charging each non-attracting subtree with mass equal to its degree. We prove that as we iterate the rational function these measures equidistribute to the equilibrium measure. Under certain conditions, notably in the case of polynomials, this reduces to the equidistribution of repelling points. We also show that repelling type II points become negligible under iteration when the Lyapunov exponent is positive. These results give a partial and a complete answer to questions of Favre and Rivera-Letelier.
