Toy model studies of tuning and typicality with an eye toward cosmology
  We investigate a number of simple toy models to explore interesting
relationships between dynamics and typicality. We start with an infinite model
that has been proposed as an illustration of how non-ergodic dynamics can
produce interesting features that are suggestive for cosmological applications.
We consider various attempts to define the infinite model more rigorously as a
limit of a finite system. None of our attempts at such rigor were able to
preserve the attractive properties. We hope our work will challenge others to
find more successful toy models. The difficulty of finding such models suggests
that connections between dynamics and typicality we hope for in cosmological
theories such as eternal inflation may not be so easy to achieve.
