A subshift of finite type with two different positive sofic entropies

Dynamical entropy is an important tool in classifying measure-preserving or topological dynamical systems up to measure or topological conjugacy. Classical dynamical entropy theory, of an action of a single transformation, has been studied since the 50s and 60s. Recently L. Bowen and Kerr-Li have introduced entropy theory for actions of sofic groups. Although a conjugacy invariant, sofic entropy in general appears to be less well-behaved than classical entropy. In particular, sofic entropy may depend on the choice of sofic approximation, although only degenerate examples have been known until now.

We present an example, inspired by hypergraph 2-colorings from statistical physics literature, of a mixing subshift of finite type with two different positive topological sofic entropies corresponding to different sofic approximations. The measure-theoretic case remains open. This is joint work with Lewis Bowen and Dylan Airey.