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罢颈迟濒别:听Ergodic theorems along trees.

Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $\{x, T(x), T^2(x), \hdots, T^n(x)\}$ in front of the point $x$. We prove a 鈥渂ackward鈥� ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over subtrees of the graph of T that are rooted at $x$ and lie behind $x$ (in the direction of $T^{-1}$). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank, where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This strengthens Bufetov鈥檚 theorem from 2000, which was the most general result in this vein. This is joint work with Jenna Zomback.

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