Thermodynamic Formalism for Countable Markov Shifts in Hyperbolic Dynamics: A Transfinite Spectral Approach

Mathematical Applications of Science Fiction We present a rigorous derivation of the thermodynamicformalism for countable Markov shifts (CMS) with a spe-cific focus on potentials of summable variation and theirtransfinite extensions in non-compact settings. By construct-ing a generalized Ruelle Transfer Operator Lϕ acting ona scale of Banach spaces adapted to the countable alpha-bet topology, we establish the existence and uniqueness ofequilibrium states (Gibbs measures) under the conditionof finite Gurevich pressure. We prove a spectral gap prop-erty for the operator using a Lasota-Yorke type inequality,thereby demonstrating exponential decay of correlations.The analysis extends to the transfinite limit of thermody-namic variables, providing a robust framework for hyperbolicdynamics on non-compact manifolds.