Modular Automorphism Groups and the Intrinsic Thermodynamics of Quasi-Crystalline C*-Algebras

This paper investigates the deep relationship between the geometric structure of aperiodic order and the thermodynamic behavior of quantum statistical mechanical systems built upon it. Focusing on C*-dynamical systems associated with quasi-lattices generated by primitive substitution rules—the mathematical foundation of quasi-crystals—we employ Tomita–Takesaki modular theory to study the equilibrium (Kubo–Martin–Schwinger, or KMS) states that arise from the natural scaling dynamics of such substitutions. Our principal result offers a full classification of the von Neumann factors obtained via the GNS representation of these KMS states, demonstrating that for a C*-dynamical system derived from a Pisot substitution with inflation factor λ > 1, the associated von Neumann algebra is a Type III_{λ^{−β}} factor, where β denotes the inverse temperature. This reveals an explicit correspondence between a geometric invariant of the aperiodic structure (λ) and a fundamental invariant in operator algebra theory (the Connes spectrum). Consequently, we show that the thermodynamic properties of these non-commutative systems are not emergent phenomena but are intrinsically and rigidly encoded in the self-similar geometry of the quasi-crystal itself.