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Gottschalk–Hedlund theorem revisited
Mathematical Research Letters ( IF 1 ) Pub Date : 2021-03-01
Xifeng Su, Philippe Thieullen

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.

中文翻译:

哥特沙克-海德隆定理再探

遍历优化和离散弱KAM理论是两个并行的理论,有几个共同的结果。例如,马瑟集是使给定可观测值的遍历平均值最小的轨道轨迹。在有利的情况下,可观察到的与其在马瑟集上的遍历最小化值是同义的,并且离散的弱KAM解起传递函数的作用。构建这样的共边界的一种可能性是使用非线性Lax–Oleinik算子。另一种可能性是通过使用折衷的同调方程。已知折现后的离散弱KAM解收敛于某些选定的弱KAM解。我们证明,在遍历优化情况下,对于最小系统上可观察到的共边界,
更新日期:2021-04-19
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