沿树的逆向遍历定理及其对自由群作用的推论
A backward ergodic theorem along trees and its consequences for free group actions
AI总结：
本文证明自由群保概率测度作用沿任意有限子树的逐点遍历定理，通过建立可数到一变换的逆向遍历定理实现，加强了Grigorchuk、Nevo-Stein的结果并证实Bufetov猜想。
摘要：
我们证明了一个新的关于自由群保概率测度（pmp）作用的逐点遍历定理，其中遍历平均是在标准凯莱图中以单位元为根、任意有限子树上取的。这一结果显著加强了Grigorchuk（1987）以及Nevo和Stein（1994）的定理，并且Bufetov在2002年曾猜想该定理的一个版本。我们的自由群定理源于一个新的——逆向的——关于可数到一pmp变换的遍历定理，其中平均是在该点的逆向轨道中任意有限高度的树上取的（即可能过去的树）。我们还讨论了该逆向定理的其他应用，特别是对带马尔可夫测度的移位映射的应用，这为自由群的边界作用给出了沿树的逐点遍历定理。
英文摘要：
We prove a new pointwise ergodic theorem for probability-measure-preserving (pmp) actions of free groups, where the ergodic averages are taken over arbitrary finite subtrees of the standard Cayley graph rooted at the identity. This result is a significant strengthening of a theorem of Grigorchuk (1987) and Nevo and Stein (1994), and a version of it was conjectured by Bufetov in 2002. Our theorem for free groups arises from a new - backward - ergodic theorem for a countable-to-one pmp transformation, where the averages are taken over arbitrary trees of finite height in the backward orbit of the point (i.e. trees of possible pasts). We also discuss other applications of this backward theorem, in particular to the shift map with Markov measures, which yields a pointwise ergodic theorem along trees for the boundary actions of free groups.
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