Anush Tserunyan : Backward and forward ergodic theorems along with 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 the forward orbit of the 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 $x$ (in the direction of $T^{-1}$). Somewhat unexpectedly, this theorem yields ("forward") ergodic theorems for group actions, in particular, one for pmp actions of free groups of finite rank 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).