Thermodynamic formalism for amenable groups and countable state spaces

Given the full shift over a countable state space on a countable amenable group, we develop its thermodynamic formalism. First, we introduce the concept of pressure and, using tiling techniques, prove its existence and further properties such as an infimum rule. Next, we extend the definitions of different notions of Gibbs measures and prove their existence and equivalence, given some regularity and normalization criteria on the potential. Finally, we provide a family of potentials that non-trivially satisfy the conditions for having this equivalence and a non-empty range of inverse temperatures where uniqueness holds.