##### Department of Mathematics,

University of California San Diego

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### Math 211 - Group Actions Seminar

## Jenna Zomback

#### University of Illinois Urbana-Champaign

## A backward ergodic theorem and its forward implications

##### 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), T2(x),..., Tn(x)} in front of the point x. We prove a "backward" 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's theorem from 2000, which was the most general result in this vein. \\ \\ This is joint work with Anush Tserunyan.

Host: Brandon Seward

### April 6, 2021

### 10:00 AM

### Zoom ID 967 4109 3409 (email Nattalie Tamam or Brandon Seward for the password)

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