A nonergodic flow rule with Galton–Watson stationary measure

A flow rule assigns to every infinite rooted tree a unit flow from the root to infinity, consistently with passage to the subtrees of the children of the root, and it defines a Markov chain on the space of trees that moves from a tree to the subtree of a child chosen according to the flow. Lyons, Pemantle and Peres showed that the chain is ergodic when a stationary measure absolutely continuous with respect to a Galton–Watson measure $\mathrm{GW}$ exists and the flow is positive on every child, and asked whether a flow rule with a stationary measure equivalent to $\mathrm{GW}$ must give an ergodic chain. It need not. For the Galton–Watson measure with offspring distribution $p_2=3/5$, $p_3=2/5$ there is a flow rule for which $\mathrm{GW}$ itself is stationary and the stationary chain has an invariant event of probability $1/2$. The rule follows a colouring of the vertices that is a measurable function of the tree, in which a vertex with two children takes the AND of their colours and a vertex with three children the OR: the flow is spread evenly over the children that have the colour of the root. The two colour classes give the ergodic decomposition of the chain.