Computable thermodynamic formalism

We investigate the theory of thermodynamic formalism from the perspective of computable analysis, with a special focus on the computability of equilibrium states. Specifically, we develop two complementary general approaches to verify the computability of equilibrium states for nonuniformly expanding computable dynamical systems. The first approach applies to dynamical systems whose topological pressure functions admit effective approximations and whose measure-theoretic entropy functions are upper semicontinuous. As a concrete application, we establish the computability of the equilibrium states for Misiurewicz-Thurston rational maps with Hölder continuous potentials. The second approach exploits prescribed Jacobians of equilibrium states through a local analysis and applies to settings where the measure-theoretic entropy functions may lack upper semicontinuity.

💡 Research Summary

This paper investigates thermodynamic formalism from the standpoint of computable analysis, focusing on the computability of equilibrium states for non‑uniformly expanding dynamical systems. The authors develop two complementary general approaches that link the computability of thermodynamic quantities—topological pressure and measure‑theoretic entropy—to the computability of the associated equilibrium measures.

The first approach, presented in Theorem 1.1, assumes that (i) the topological pressure function can be effectively approximated by a uniformly computable family of potentials whose pressure values form an upper‑semi‑computable real sequence, (ii) the pressure of the target potential is lower‑semi‑computable, and (iii) the entropy map ν↦h_ν(T) is upper semicontinuous on the space of invariant measures. Under these hypotheses, and assuming uniqueness of the equilibrium state for each system in a uniformly computable family, the equilibrium measures form a uniformly computable sequence. The key idea is to use the set of tangent functionals to the pressure function; these tangents correspond bijectively to equilibrium states, allowing one to recover the measure from effective pressure approximations. As a concrete illustration, Theorem 1.2 applies this framework to computable Misiurewicz–Thurston rational maps with Hölder continuous potentials, establishing the uniform computability of their unique equilibrium states.

The second approach, embodied in Theorem 1.3, does not require upper semicontinuity of the entropy map. Instead, it exploits prescribed Jacobians of equilibrium states. For a compact metric space (X,ρ) and a Borel map T, the authors define admissible subsets and a set M(X,T;Y,J) of measures satisfying a Jacobian inequality (1.2) on all admissible subsets of an open set Y. Assuming the existence of recursively enumerable families of open admissible sets {Y_{n,k}} and lower‑semi‑computable Jacobians {J_n}, together with recursively compact sets K_n that intersect M(X,T_n;Y_n,J_n) in a single measure μ_n, the theorem guarantees that the sequence {μ_n} is uniformly computable. This method is particularly suited to systems where entropy may be discontinuous, such as those with periodic critical points.

The authors apply the second method to expanding Thurston maps. Theorem 1.4 shows that for a computable expanding Thurston map on the Riemann sphere with computable critical points, the measure of maximal entropy is computable. The proof constructs suitable Jacobians and recursively compact approximations of the invariant measure, bypassing the need for entropy semicontinuity.

Beyond these main results, the paper provides extensive background on computable metric spaces, semi‑computable open sets, and the computable structure on the space of Borel probability measures. It also discusses the relationship between the two approaches, noting that the first is global and relies on convex analysis of the pressure functional, while the second is local and leverages transfer operators and Jacobian estimates. The work extends earlier results on the computability of Brolin–Lyubich measures and of invariant measures for uniformly expanding systems, demonstrating that statistical computability can coexist with significant dynamical complexity even in non‑uniformly hyperbolic regimes.

In the concluding sections, the authors outline future directions, including extending the framework to multi‑potential settings, handling systems with multiple equilibrium states, and developing concrete algorithms for practical computation of equilibrium measures in complex dynamics. Overall, the paper makes a substantial contribution by establishing robust, algorithmic foundations for thermodynamic formalism in a broad class of dynamical systems.