Anush Tserunyan (McGill University)

Backward ergodic theorem along trees and its consequences

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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 $\{x, T(x), T^2(x), ..., T^n(x)\}$ in the "future" of a point $x$. In joint work with Jenna Zomback, we prove a backward ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over arbitrary trees of possible "pasts" of $x$. Somewhat unexpectedly, this theorem yields ergodic theorems for actions of free groups, where the averages are taken along arbitrary subtrees of the standard Cayley graph rooted at the identity. This strengthens results of Grigorchuk (1987), Nevo (1994), and Bufetov (2000).