# Dynamics from iterated averaging

## TL;DR

This paper proves that the closure of the semigroup generated by conditional expectations on a Lebesgue space includes all measure-preserving automorphisms, linking ergodic theory with a water tank equilibration puzzle.

## Contribution

It establishes a connection between iterated averaging operators and measure-preserving transformations, providing a new perspective on their closure properties.

## Key findings

- The strong operator closure contains all measure-preserving automorphisms.
- The result is based on solving a water tank equilibration puzzle.
- Provides a novel link between ergodic theory and a physical analogy.

## Abstract

We prove that for a standard Lebesgue space $X$, the strong operator closure of the semigroup generated by conditional expectations on $L^\infty(X)$ contains the group of measure-preserving automorphisms. This is based on a solution to the following puzzle: given $n$ full water tanks, each containing one unit of water, and $n$ empty ones, how much water can be transferred from the full tanks to the empty ones by repeatedly equilibrating the water levels between pairs of tanks?