折扣Gottschalk-Hedlund定理
Discounted Gottschalk-Hedlund theorem
摘要：
遍历优化和离散弱KAM理论是两个并行的理论，有一些共同的结果。例如，马瑟集是使给定可观测量的遍历平均最小化的轨道的轨迹。在较好的情况下，观测量与其在Mather集上的遍历极小值是同调的，离散的弱KAM解起着传递函数的作用。一种可能的建设这样的共边界是通过使用非线性Lax-Oleinik算子。另一种可能性是通过使用折扣上同调方程。已知折扣离散弱KAM解收敛于某个选定的弱KAM解。我们证明了，在遍历优化的情况下，一个最小的系统上的共界观测，折扣传递函数收敛当且仅当观察是平衡的。
英文摘要：
Ergodic optimization and discrete weak KAM theory are two parallel theories with several results in common. For instance, the Mather set is the locus of orbits which minimize the ergodic averages of a given observable. In the favorable cases, the observable is cohomologous to its ergodic minimizing value on the Mather set, and the discrete weak KAM solution plays the role of the transfer function. One possibility of construction of such a coboundary is by using the non linear Lax-Oleinik operator. The other possibility is by using a discounted cohomological equation. It is known that the discounted discrete weak KAM solution converges to some selected weak KAM solution. We show that, in the ergodic optimization case for a coboundary observable over a minimal system, the discounted transfer function converges if and only if the observable is balanced.
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