Multifractals, Mumford curves, and Eternal Inflation

We relate the Eternal Symmetree model of Harlow, Shenker, Stanford, and Susskind to constructions of stochastic processes arising from quantum statistical mechanical systems on Cuntz--Krieger algebras. We extend the eternal inflation model from the Bruhat--Tits tree to quotients by p-adic Schottky groups, again using quantum statistical mechanics on graph algebras.