Evolution with state-dependent mutations

Recent evolutionary models have introduced "small mutation rates" as a way of refining predictions of long-run behavior. We show that this refinement effect can only be obtained by restrictions on how the magnitude of the effect of mutation on evolution varies across states of the system. In particular, given any model of the effect of mutations, any invariant distribution of the "mutationless" process is close to an invariant distribution of the process with appropriately chosen small mutation rates.