Thermodynamical Formalism for a Modified Shift Map

We introduce a transfer operator and use it to prove some theorems of a classical flavor from thermodynamical formalism (including existence and uniqueness of appropriately defined Gibbs states and equilibrium states for potential functions satisfying Dini’s condition and stochastic laws for Hölder continuous potential and observable functions) in a novel setting: the “alphabet” E is a compact metric space equipped with an apriori probability measure ν and an endomorphism T . The “modified shift map” S is defined on the product space EN by the rule (x1x2x3 . . . ) 7→ (T (x2)x3 . . . ). The greatest novelty is found in the variational principle, where a term must be added to the entropy to reflect the transformation of the first coordinate by T after shifting. Our motivation is that this system, in its full generality, cannot be treated by the existing methods of either rigorous statistical mechanics of lattice gases (where only the true shift action is used) or dynamical systems theory (where the apriori measure is always implicitly taken to be the counting measure).