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arXiv · 2104.02635

Quantitative ergodic theorems for actions of groups of polynomial growth

Guixiang Hong·Wei Liu
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Abstract

We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As a consequence, we deduce in this setting the quantitative ergodic theorem, in particular, the upcrossing inequalities with exponential decay. The ideas or techniques involve probability theory, non-doubling Calder\'on-Zygmund theory, almost orthogonality argument and some delicate geometric argument involving the balls and the cubes on the group equipped with a not necessarily doubling measure.

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Guixiang Hong, Wei Liu. 2021-04-06. Quantitative ergodic theorems for actions of groups of polynomial growth. https://doi.org/10.1017/etds.2025.10210

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