Equilibrium states beyond specification and the Bowen property

TL;DR

This paper establishes conditions for the uniqueness of equilibrium states in symbolic dynamical systems lacking the classical specification and Bowen properties, broadening the scope of thermodynamic formalism.

Contribution

It introduces a new approach that handles systems without specification and Bowen properties by controlling obstructions, avoiding inducing schemes and Perron-Frobenius operators.

Findings

Unique equilibrium states exist under weakened conditions.

Potential functions with unique states include those beyond Bowen property.

Applications to interval maps with indifferent fixed points and non-Markov structures.

Abstract

It is well-known that for expansive maps and continuous potential functions, the specification property (for the map) and the Bowen property (for the potential) together imply the existence of a unique equilibrium state. We consider symbolic spaces that may not have specification, and potentials that may not have the Bowen property, and give conditions under which uniqueness of the equilibrium state can still be deduced. Our approach is to ask that the collection of cylinders which are obstructions to the specification property or the Bowen property is small in an appropriate quantitative sense. This allows us to construct an ergodic equilibrium state with a weak Gibbs property, which we then use to prove uniqueness. We do not use inducing schemes or the Perron--Frobenius operator, and we strengthen some previous results obtained using these approaches. In particular, we consider $\beta$-shifts and show that the class of potential functions with unique equilibrium states strictly contains the set of potentials with the Bowen property. We give applications to piecewise monotonic interval maps, including the family of geometric potentials for examples which have both indifferent fixed points and a non-Markov structure.