Strong law of large numbers for Bernoulli sequences and Gibbs measures on subshifts for finite and infinite alphabets - Borsato, Luísa Bürgel (2022)
We prove a characterization of the Strong Law of Large Numbers for Bernoulli sequences [BHS20] in terms of the concept of a product disintegration, which is a generalization of the notion of exchangeability. We also study properties of Gibbs measures ([Geo11, Kel98, Cap76, Rue04, LN08]) in different contexts. For finite state spaces over countable groups, we prove the equivalence between measures satisfying the DLR equations and conformal measures [BM21], based on the work of [Kim15]. Still in the finite state space framework, but restricted to one-dimensional irreducible sofic shifts, we prove a version of the Dobrushin-Lanford-Ruelle theorem ([Dob68, LR69]), that connects DLR measures to equilibrium measures ([BM20]). At last, in the infinite alphabet case in a countable amenable group, we prove the existence of the pressure, which we define through a limit over sets that are becoming more and more invariant and that it satisfies a infimum rule. Moreover, we prove that pressure can be obtained as the supremum of the pressures associated to finite alphabet subsystems. Also, based on the work of [Mui11b, Kel98], we prove that the notions of DLR, conformal, Bowen-Gibbs and equilibrium measures for exp-summable potentials with summable variations are connected and are equivalent given some conditions. Finally, we prove a version of the Dobrushins uniqueness theorem for this setting.