Title: Equilibrium States from Symbolic Dynamics to Quantum Spin Systems
Abstract: The Gibbs measure of a Hölder potential on a subshift of finite type is characterized via Ruelle’s Perron-Frobenius theorem: equilibrium corresponds to an eigenmeasure of the transfer operator, while its spectral gap governs decay of correlations and mixing.
We explain how this picture extends beyond the commutative setting: the KMS condition provides a general notion of equilibrium for C*-dynamical systems, which in the symbolic setting recovers the classical eigenmeasure structure underlying Gibbs states.
Passing to the noncommutative framework yields the appropriate formulation of equilibrium for quantum lattice spin systems, where the space of KMS states admits an ergodic decomposition mirroring the classical theory.
We conclude by discussing a quantum analogue of the Martinelli-Stroock-Zegarlinski program: under what conditions does uniqueness of the KMS state imply rapid mixing of the associated quantum dynamics?