LAUSR

Abstract We consider a family of ⁎-commuting local homeomorphisms on a compact space, and build a compactly aligned product system of Hilbert bimodules. The Nica–Toeplitz algebra of this system carries a gauge action of a higher-dimensional torus, and there are many possible dynamics obtained by composing with different embeddings of the real line in this torus. We study the KMS states of these dynamics. For large inverse temperatures including ∞, we describe the simplex of KMS states on the Nica–Toeplitz algebra. We illustrate our main theorem by considering backward shifts on the infinite-path spaces of a class of k-graphs whose shift maps ⁎-commute.