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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.