AI 中文总结
本文针对紧可度量化阿贝尔群上的连续满同态，显式构造连续函数使哈尔测度成为其唯一不变最大化测度，该函数为三角多项式递归序列的一致极限，特定情形下可得到可精确计算的傅里叶展开。
AI 中文摘要
我们解决了如下问题：显式构造一个连续函数，使得加倍映射的唯一最大化测度为勒贝格测度。更一般地，给定一个非平凡的紧可度量化阿贝尔群，以及一个归一化哈尔测度为遍历的连续满同态，我们显式构造了该群上的一个连续函数，使得哈尔测度是其唯一的不变最大化测度。该函数是具有有理系数的三角多项式递归定义序列的一致极限；递归的每个参数都由闭式公式给出，每一步都是精确的，收敛速度是显式的。在特定情形下，我们还得到了按经典频率顺序的一致收敛傅里叶展开，其每个系数都是有理数，且可通过有限过程精确计算。
英文摘要
We solve the problem of explicitly constructing a continuous function whose unique maximizing measure for the doubling map is Lebesgue measure. More generally, given a nontrivial compact metrizable abelian group and a continuous surjective endomorphism for which normalised Haar measure is ergodic, we explicitly construct a continuous function on the group for which Haar measure is the unique invariant maximizing measure. The function is the uniform limit of a recursively defined sequence of trigonometric polynomials with rational coefficients; every parameter of the recursion is given by a closed formula, every step is exact, and the rate of convergence is explicit. In specific cases, we further obtain a uniformly convergent Fourier expansion in the classical frequency order, each of whose coefficients is rational and computable exactly, by a finite procedure.
Comments19 pages
