Generalizing the uniqueness of equilibrium states in a conditional setting

Bowen and Sinai introduced many tools which enabled them to investigate properties of equilibrium states on systems beyond full shifts: Anosov systems and Axiom A systems. Since then people have been trying to generalize the theory in many directions such as certain classes of Z-actions or a relative setting. In a relative setting, we are concerned with a dynamical system (X,T ) equipped with a distinguished factor map π : (X,T ) → (Y, S) and we ask which invariant measures μ ∈ MT (X) maximize hμ(T ) + ∫ V dμ among the measures whose projection on (Y, S) is fixed to be ν ∈MS(Y ), where V ∈ C(X) is usually assumed to be Hölder-continuous. Such measures, called relative equilibrium states, are not necessarily unique even when both (X,T ) and (Y, S) are full shifts. Nonetheless, Petersen, Quas, and Shin showed that there are only finitely many such ergodic measures if V = 0 and (X,T ) and (Y, S) are transitive subshifts of finite type [3]. Recent progress in this area is as follows. Allahbakhshi and Quas strengthened the finiteness result by giving a specific bound cπ related to the structure of π [2]. Allahbakhshi, Antonioli, and Yoo generalized it by dropping the condition V = 0 [1]. Yoo proved that there are exactly cπ such measures, counting multiplicities [4].