# Logic Seminar

In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation TT, one takes averages of a given integrable function over the intervals {x,T(x),T2(x),…,Tn(x)}{x,T(x),T2(x),…,Tn(x)} in front of the point xx. We prove a "backward" ergodic theorem for a countable-to-one pmp TT, where the averages are taken over subtrees of the graph of TT that are rooted at xx and lie behind xx (in the direction of T−1T−1). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, for pmp actions of finitely generated groups, where the averages are taken along set-theoretic (but backward) trees on the generating set. This strengthens Bufetov's theorem from 2000, which was the leading result in this vein. This is joint work with Jenna Zomback.