Strong approximations of shifts and the characteristic measures problem

Every symbolic system supports a Borel measure that is invariant under the shift, but it is not known if every such systems supports a measure that is invariant under all of its automorphisms, known as a characteristic measure. We give sufficient conditions to find a characteristic measure, additionally showing when it can be taken to be a measure of maximal entropy. The class of systems to which these sufficient conditions apply is large, containing a dense \(G_{\delta }\) $G_{\delta }$ set in the space of all shifts on a given alphabet, and is also large in the sense that it is closed under taking factors. We also investigate natural systems to which these sufficient conditions apply.